2017 Winner

## Starry Reckoning: Reference and Analysis in Mathematics and Cosmology

by### Emily Rolfe Grosholz

#### About the book

This book is a further development of the ideas in two earlier books by Emily Rolfe Grosholz. Her 1991 book: *Cartesian Method and the Problem of Reduction* is based on the notion that ideas can develop through bringing together two, formally separated, domains. The obvious example is Descartes’ development of analytic geometry through bringing together classical geometry and algebra. Her 2007 book: *Representation and Productive Ambiguity in Mathematics and the Sciences* introduces another quite novel idea, namely that mathematical systems are always ambiguous, and that this ambiguity helps with the development of new mathematical ideas. The present book is the culmination of her research in this field. It introduces the key notions of analysis and reference, and develops them in a way, which makes use of her earlier ideas.

Grosholz’s thesis is that mathematics and science require both discourses of analysis, and discourses of *reference*. These two types of discourse have to brought together, and this conjunction is fruitful for the development of knowledge in the way she showed in her first book. The same term often occurs in both types of discourse, and thus becomes to some degree ambiguous. However, this is the kind of *productive ambiguity* which she discussed in her second book. When Grosholz speaks of ‘analysis’, this might be interpreted as referring to analysis as it is used in analytic philosophy, i.e. logical analysis. In fact Emily Rolfe Grosholz uses analysis, in a different sense which she takes from Leibniz, namely as the search for the conditions of intelligibility of a notion. One interesting point here, which Grosholz discusses on p. 41, is that both Russell and Couturat hailed Leibniz as the founder of formal logic. However, Grosholz interprets Leibniz in a broader sense. Grosholz’s notion of analysis leads to an interesting discussion of the status and importance of mathematical logic. She discusses the views of Carlo Cellucci who has argued that mathematical logic failed in its foundational aims, and is therefore an area, which should be abandoned. She agrees with him partly, but not entirely. She says (p. vi): “… mathematical logic … is not an over-discourse that should supplant the others”, but then significantly adds (p. vi): “but one of many, which can be integrated with other mathematical discourses in a variety of fruitful ways.” Thus mathematical logic, even though it failed in its original foundational aims, is still a valuable branch of mathematics. This point of view is illustrated by some of her cases studies.

One of the most striking of the case studies is the analysis of Wiles’ recent proof of Fermat’s Last Theorem. She prefaces her treatment of Wiles by a most interesting discussion of Gödel’s incompleteness theorems. She says (pp. 88-9):

*“Gödel … to carry out his proof … must use modes of representation that lend themselves to logical analysis (Russell’s notation) but not to computing or referring, and other modes of representation that lend themselves to successful reference (Indo-Arabic/Cartesian notation). He must use disparate registers of the formal languages available to him, combine them, and exploit their ambiguity.”*

This is a marvellous account of Gödel’s proof which uses Grosholz’s key notions of analysis, reference, and productive ambiguity. Going on from this, to her discussion of Wiles’ proof, she points out that the proof has been extensively analysed by mathematical logicians, notably Angus Macintyre. Her point is that Macintyre is interested in different questions from Wiles. Macintyre is, for instance, interested in whether the proof can be carried out within 1st order Peano arithmetic. This introduces a new discourse which is different from Wiles’ discourse which involves integers, rational numbers, modular forms and elliptic curves. The interaction of this new discourse with the old one may give rise to progress. As she says (p. 101): “the interaction between logic and number theory … may give rise to novel objects, procedures and methods still to be discovered.”

Grosholz goes on to apply her approach to the representation of time, covering first Galileo, Newton and Leibniz, and then the period from 1700 to the present, which brings in the question of thermodynamics and the direction of time. In this area, the discourse of analysis is connected with arithmetic and geometry, while that of reference with clocks (see pp. 133-4).

The final chapter moves definitely into the region of physics by considering astronomical systems. Once again Grosholz applies her analysis/reference approach, and considers a broad historical span, starting with Kepler and Newton, continuing through the 19th century, and ending in the 1970s with the enigma of the Andromeda galaxy, and Vera Rubin’s consequent postulation of dark matter.

Springer

ISBN 978-3-319-46689-7

202 pages · 2016

#### About the author

Emily Rolfe Grosholz is Edwin Erle Sparks Professor of Philosophy, African American Studies and English, and a member of the Center for Fundamental Theory / Institute for Gravitation and the Cosmos at the Pennsylvania State University. She earned her Bachelor of Arts degree at the University of Chicago, and her Ph. D. at Yale University. From 1979 to the present, she has taught in the philosophy department at the Pennsylvania State University, interspersed with various research or teaching appointments at the University of Toronto, the University of Pennsylvania, the National Humanities Center, the Leibniz Archives in Hannover, Germany, Clare Hall / Cambridge University, and the University of Paris.

She is a Chercheur Associe Etranger of SPHERE Université Paris Denis Diderot (Paris 7) and UMR 7219 Centre National de la Recherche Scientifique; Member, Centre d’Etudes Leibniziennes (Mathesis), La Sorbonne; Life Member, Clare Hall, University of Cambridge; and Associate, Center for Philosophy of Science, University of Pittsburgh.

She is a member of the editorial board of *Studia Leibnitiana*, the *Journal of Humanistic Mathematics*, the *Journal of Mathematics and the Arts*, and *The Hudson Review*; and a member of the board of directors of the *Journal of the History of Ideas*, and the steering committee of the *Association for the Philosophy of Mathematical Practice*. Her monographs include *Cartesian Method and the Problem of Reduction* (Oxford University Press 1991), *Leibniz’s Science of the Rational* (Frantz Steiner Verlag, 1998), *Representation and Productive Ambiguity in Mathematics and the Sciences* (Oxford University Press 2007), and *Starry Reckoning: Reference and Analysis in Mathematics and Cosmology* (Springer 2016), which was awarded the Fernando Gil International Prize for philosophy of science. She has also edited or co-edited six collections of essays, and written eight books of poetry, most recently The *Stars of Earth: New and Selected Poems* (Word Galaxy Press, 2017). Her book *Great Circles: The Transits of Mathematics and Poetry*, which combines her philosophical and literary work, is due out in 2018 from Springer.

### The 2017 Fernando Gil Prize for Philosophy of Science

In this book Emily Rolfe Grosholz adopts the approach of history and philosophy of science and mathematics, and indeed defends this approach in the course of the book. On the philosophical side, Emily Rolfe Grosholz develops a clear and original point of view. This is that mathematics and science require both discourses of analysis and discourses of reference. Here ‘analysis’ does not mean ‘logical analysis’, but has a sense which Emily Rolfe Grosholz takes from Leibniz, namely: ‘the search for conditions of intelligibility’. Her views therefore are a development of some Leibnizian notions, and her work gives an interesting new interpretation of Leibniz. Her philosophical thesis is illustrated by a great variety of historical case studies, which include cases from the early modern period, the 19th century, and recent research. There are also examples from both mathematics and physics (cosmology). The successful application of the underlying philosophical thesis to so many examples shows that it is both plausible and fruitful, and the case studies themselves are very interesting. The jury was particularly impressed by Emily Rolfe Grosholz’ study of Wiles recent proof of Fermat’s last theorem. They regarded it as admirable that a philosopher of mathematics should reflect on a recent and technically very difficult proof. Such a strategy can produce an ‘immanent’ philosophy. A further study of McIntyre’s logical investigation of Wiles’ proof leads Emily Rolfe Grosholz to suggest a new attitude to mathematical logic as a discipline. In her own words (p.vi): “ … mathematical logic … is not an over-discourse that should supplant others … but one of many, which can be integrated with other mathematical discourses in a variety of fruitful ways.” Given all these impressive features of Emily Rolfe Grosholz’ book, the jury judged it to be a worthy winner of the Fernando Gil prize for 2017.