Emily Rolfe Grosholz

[Silvia De Toffoli]

Dear All,

It is with great sadness that I announce the passing of Emily Rolfe Grosholz, Professor of Philosophy, English, and African American Studies at Pennsylvania State University.

Emily has been active in our community and served on the APMP Directive Committee from 2014 to 2020. She is well known for her work in philosophy of mathematics and poetry. She is the author of several books, including Cartesian Method and the Problem of Reduction (1991), Representation and Productive Ambiguity in Mathematics and the Sciences (2007), and Starry Reckoning: Reference and Analysis in Mathematics and Cosmology (2016).

Carlo Cellucci has kindly shared the following remembrance of Emily and her work:

Emily Rolfe Grosholz, a prominent member of the APMP, passed away on May 2 at the age of 76, surrounded by the affection of her husband, Robert R. Edwards, and her children.

Emily is a nearly unique case of excellence in both philosophy, particularly philosophy of mathematics, and poetry, as witnessed by the numerous valuable books she published in both fields and by the international recognition of her value.

The website https://emilygrosholz.com/ contains quite rich information about Emily’s academic and literary career and publications. In this note I will provide a summary of her contribution to the philosophy of mathematics.

Emily is opposed to the vision of the foundational schools, of a unified mathematics through its reduction to a single mathematical theory. In her view, this vision is not only actually unattainable but even misleading as a regulative ideal. For, mathematics is a set of rationally related but autonomous fields, rather than a unity manque. 

She argues that some important areas of mathematical activity do not even lend themselves to axiomatization, they exist rather as a set of solved and unsolved problems concerning certain kinds of mathematical objects. Yet, they can come to stand in significant relations to other areas of mathematical activity that may themselves be powerful vehicles for solving problems and constructing novel objects. 

In Emily’s view, these relations can lead to the creation of hybrids, that is, objects that exhibit features of different areas of mathematics. Their multivalence endows them with their characteristic manageable inconsistency and suggestiveness. Hybrids take on a life of their own, and become crucial to the growth of knowledge, because they can stimulate discoveries in unexpected ways. They can even lead to the emergence of new areas of mathematics, growing out of the hypotheses posed with respect to the hybrids. An example is Descartes’s introduction of curves as geometric-algebraic objects, which led to the rise of analytic geometry.

Another source of mathematical knowledge is what Emily calls ‘productive ambiguity’. Different representations reveal different aspects of intelligible things, the problems in which they occur, and the procedures that successfully solve them. When distinct representations are juxtaposed and superimposed, the result is often a productive ambiguity that expresses and generates new knowledge. An example are the diagrams in Newton’s Principia. If one reads them as consisting of finite line segments and areas, Euclidean theorems can be applied to solve geometry problems. But by reading Newton’s diagrams as comprising infinitesimal quantities, these solutions can be applied to issues of motion and force. 

While being opposed to the vision of the foundational schools, however, Emily is also opposed to the anti-foundational idea that mathematics is not about truth but only about plausibility, and is therefore always subject to revision. In her view, the idea that we should focus on plausibility rather than truth, and construe justification as always somewhat empirical and subject to revision, fails to capture the peculiar nature of mathematical reasoning. Mathematics essentially differs from the natural sciences because its propositions are true and the things of mathematics are determinate and unchanging as the things of nature are not. Mathematics is inexhaustible not because it is a set of propositions that are merely plausible and therefore always subject to revision, but because the things of mathematics are infinitely analyzable, since we can always discover new aspects of them when they are brought into relation with other mathematical things.

Furthermore, Emily tends to play down Lakatos’ basic idea in his book Proofs and Refutations, that the philosophy of mathematics should primarily concern itself with the logic of mathematical discovery. 

On the other hand, however, Emily praises Imre Lakatos’s book for applying the historical approach to the philosophy of mathematics. The historical approach was already well-established in the philosophy of science, but was entirely alien to the foundational schools, and therefore was entirely new in the philosophy of mathematics. Lakatos’s book inspired many young philosophers to think about mathematics in a new way.

In Emily’s view, rather than trying to state rules for a logic of discovery, efforts should be spent on studying the history of mathematics. In this respect, it is very encouraging that the philosophy of mathematics in the early twenty-first century is undergoing a long-awaited and long-overdue sea change. Philosophers of mathematics are turning to a serious study of the history of mathematics, they are trying to give an account of how mathematical knowledge grows and how problems are solved. 

This is only a summary of Emily’s contributions to the philosophy of mathematics. But I cannot end without saying that many colleagues and scholars will miss Emily’s intellectual curiosity and research rigor, and her many friends will especially miss her great humanity.

Carlo Cellucci

Best wishes,

Silvia

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Silvia De Toffoli
www.silviadetoffoli.net