*Circles Disturbed*are a diverse selection. The contributors include philosophers, historians of science, mathematicians, and literary theorists, among others, and their offerings range from broad, high level surveys to narrow studies of specific topics. Most of them make no assumptions about mathematical knowledge, or no more than a high school student might have.

There were clearly other possibilities, but the order of presentation chosen by Doxiadis and Mazur works well. First come the essays that are predominantly historical in focus, then the more philosophical ones, and finally the more literary ones.

Amir Alexander's "From Voyagers to Martyrs: Towards a Storied History of Mathematics" is a potted history of some narrative themes or tropes in the early modern history of mathematics: in the 17th century, links to ideas about exploration and discovery; in the 18th, the idea of the mathematician as a "natural man", with d'Alembert as an exemplar; and, in the 19th, the mathematician as martyr, epitomised by figures such as Galois and Abel. Alexander's subject is not the lives of these figures but the stories told about them, which have often not been terribly historical.

Peter Galison's "Structure of Crystal, Bucket of Dust" is a study of
American physicist John Archibald Wheeler, focusing on the influence of
machines in his life and thought, from his childhood to the presentation
of the mathematics in his magnum opus *Gravitation*. This is contrasted
with the approach of the French collective mathematician Bourbaki.

Federica la Nave presents an account of Bombelli and his introduction of imaginary numbers. Here the survival of manuscripts that were later reworked allows us to see how his ideas changed over time.

In "Hilbert on Theology and Its Discontents" Colin Mclarty probes one of the origin myths of modern mathematics, in which Paul Gordan is supposed to have said, of a proof of Hilbert's, "this is not mathematics, it is theology". The reality is much more complex, and sheds light on the work of Emmy Noether, who "was in the most obvious sense a joint heir of Gordan and Hilbert".

Though centred on an analysis of a paper by Thomason and Trobaugh, "Higher Algebraic K-Theory of Schemes and of Derived Categories", Michael Harris's long piece "Do Androids Prove Theorems in Their Sleep?" is much broader. It touches on the different genres of mathematical writing, the role of narrative in proofs (illustrated by the rewriting of a key lemma as a romance), the possibility of automated proving systems, the challenges an android might face in doing mathematics or in communicating its results to human mathematicians, and the notion of a "key" in a proof or its discovery.

Barry Mazur looks at Kronecker's *Jugendtraum*, a vision of his youth
that became Hilbert's twelfth problem, exploring how it fits into the
history of algebraic number theory and how it illustrates concepts of
explanation and explicitness and the contrast between intuition or vision
and "proceeding step by step". This is one of the essays where one needs
to be able to follow the mathematics to fully appreciate the argument,
though an understanding of simple complex algebra will suffice.

The difference between showing and telling is a commonplace of creative writing advice, but a similar divide exists in mathematics between formal definition and illustration through examples. In "Vividness in Mathematics and Narrative", Timothy Gowers explores the difference between these approaches , using examples from group theory and the calculation of highest common factors, along with excerpts from literary texts.

Bernard Teissier's "Mathematics and Narrative: Why are Stories and Proofs Interesting?" hints at much, attempting among other things to invoke vestibular and visual physiology as a basis for the real line, but provides little substance. This was, perhaps, a little too "French" for my liking.

In "Narrative and The Rationality of Mathematical Practice" David Corfield applies to mathematics the epistemological theory of Alasdair MacIntyre. Drawing on Aquinas and Aristotle, this suggests viewing mathematics as "a craft tradition" as an alternative to ahistorical "encyclopedic" and relativist "genealogical" approaches.

Apostolos Doxiadis's "A Streetcar Named Proof" is the longest piece in the volume, taking up over a hundred pages. It traces the origins of ancient Greek mathematical proof in narrative, in poetic storytelling but above all in forensic rhetoric, with a key role played by chiasmus and ring-composition. Doxiadis goes into quite detailed analysis of example proofs and literary texts, making the result quite dense and perhaps of specialist interest.

G.E.R. Lloyd presents "An Aristotelian Perspective" on some proofs of
Euclid, analysing them as sequential steps, involving *energeia*, or
"the actualization of a certain potentiality".

In "Adventures of the Diagonal: Non-Euclidean Mathematics and Narrative" Arkady Plotinsky erects a broad, sweeping dichotomy between the "Euclidean" and the "non-Euclidean", the latter of which he extends far beyond geometry, to include irrational numbers, quantum mechanics and the modernist novel, among other things. The Argand plane, he argues, is not an intuitive representation of complex numbers, which as a result are "non-Euclidean": "unlike real numbers or vectors, complex numbers as such cannot be assigned lengths or be ordered, and hence they cannot be used in measurements". This seems way too simplistic a categorisation to be at all useful.

David Herman presents some diagrams illustrating different kinds of focalisation theory. As "Formal Models in Narrative Analysis" these are pretty weak, failing to convince that metanarratology is in any state for useful formalisation.

In "A Narratological Perspective", Uri Margolin surveys six areas of contact between mathematics and narrative. Five of these are only glanced over: literary portrayals of mathematicians, use of codes or riddles, use of mathematical methods to create formal structure (as practised by the Oulipo), or use of concepts of infinity, recursion, and so forth, and use of mathematical concepts in theories of narrative. The focus is on a number of concepts which are "similar or analagous" in the two domains, such as freedom of invention and choice of constraints, the ontological status of fictional characters and mathematical objects, truth criteria, sequence and levels of hierarchy, simulation and speculation about the future, and decision-making.

And finally Jan Christoph Meister's "Tales of Contingency, Contingencies of Telling" looks at narrative subjectivity and what we can learn by sketching in outline the design of a Story Generating Algorithm.

As with all such collections, *Circles Disturbed* is a mixed bag.
Those involved with the history and philosophy of mathematics should find
the majority of the pieces relevant; those with interests primarily in
narrative fewer. The general reader, just browsing, might want to start
with the pieces by Harris, Gower, and Margolin.

November 2012

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