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Freedom in Mathematics

Pierre Cartier, Jean Dhombres, Gerhard Heinzmann, and Cédric Villani
Publication Date: 
Number of Pages: 
[Reviewed by
Darren Glass
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When I applied for college several decades ago, I remember that one school asked applicants to write an essay in which they would choose several famous people to invite over for a fantasy dinner party. There are any number of ways to approach such a question — some people might choose a group that has wildly varied backgrounds and interests to see what they would discuss, while other people would prefer to gather a group of the world’s leading experts in a given area to do a deep dive into a given topic. The French publishing house La Ville Brûle has taken the latter approach with the books in their Collection: 360 series, in which they gather a handful of experts in a given scientific discipline for a free-wheeling conversation. As (my bad translation of) the publisher’s website describes:

By asking questions about science from different viewpoints, the opportunity is given to the reader to question the role of science in society. The form adopted by these works moves away from the classic written monologue to give way to dialogue and enter into a dynamic of debates, opinions, convictions, while highlighting the modes of production of science.

For the francophones:

En posant sur les questions de sciences des regards croisés, l’occasion est donnée au lecteur de s’interroger sur la place des sciences dans la société. La forme adoptée par ces ouvrages s’éloigne du classique monologue écrit pour laisser la place au dialogue et entrer dans une dynamique de débats, d’opinions, de convictions, tout en mettant en évidence les modes de production de la science.

Books in this collection include titles on climate change, biodiversity, multiverses, and more. For one such volume, entitled Mathématiques en Liberté, they brought together Pierre Cartier, Jean Dhombres, Gerhard Heinzmann, and Cédric Villani for a wide-ranging discussion about the nature of mathematics and its relationship with other sciences, with education, and with reality itself. Springer has now released this book in an English translation, and it is a very engaging read.

Reading this book allows one to feel like a fly on the wall of a parlor where these four great minds are discussing a number of different issues related to mathematics. Two of the names will probably be very familiar to readers of MAA Reviews: Cartier is one of the founders of modern algebraic geometry and a key member of the Bourbaki group, and Villani won the Fields Medal in 2010 for his work in differential geometry and related fields. The other two names are likely less familiar (at least they were to this reviewer), but both of significant credentials: Dhombres is a historian of mathematics who has worked on issues of the epistemology of mathematics for the last half century, and Heinzmann is a German philosopher who works on issues of intuition in general and has particular interests in the work of Poincaré.

The conversation, which is moderated by the journalist Sylvestre Huet, is divided into chapters entitled “On The Origin of Mathematics”, “Mathematics and Reality”, “Political and Social History of Mathematics Education”, and “The Nature and Challenges of Mathematical Research”, which are then each divided further into sections, although the conversation involves much back and forth and covers lots of ground. Most of the sections begin with Huet posing a question (examples include “Is it still thought today that nature is written in mathematical language (Galileo) or that there is a ‘pre-established harmony’ (Leibniz) between mathematics and reality” and “how is [mathematical research] experienced at individual level, as regards motivations, rhythm, results. but also as regards needs and tools?”) and then the four panelists go back and forth discussing it, and often moving completely away from the original question.

The panelists are opinionated and at times blunt, and they have no hesitation about contradicting one another on certain points (whether mathematics can be successfully done in solitude, for example, or whether it is best to think of mathematics as a tool or a language or something else altogether). At the same time, they are all very intelligent and have a deep background, and have clearly spent time thinking about these issues, so the conversation does not seem completely off-the-cuff.

Often, when listening in to a group of people who I don’t know, I find myself missing certain references, and the same is true with this conversation. Segments of the book discuss math education or politics in France, and the conversation does not slow down to fill in details for those of us who are not as familiar with that country — if you don’t know the differences between mathematicians working at CNRS and IUF and IHES then you are likely to have trouble following parts of the conversation. Similarly, they often refer in passing to the work of other mathematicians, and while the editors give very brief biographical information about these mathematicians in footnotes, there were still a number of names dropped that I was not familiar with. There were times when this was frustrating, but it clearly was a choice that allowed the editors to capture the conversation in “real time” rather than having to heavily annotate the conversation. It is also worth noting that there are a number of places where the language was a bit stilted and might have benefitted from a smoother translation, but this did not detract from the enjoyment of reading their thoughts.

In addition to the text of the conversation itself, the book contains biographical sketches of the four participants and reading lists that they each suggest for the reader who wants to think more about the topics. And the four of them give more than enough to think about as they discuss a number of deep and fascinating topics. The book is brief, and certainly reads more like an extended version of the kinds of interviews that one reads in the Notices of the American Mathematical Society than it does a deep and heavily researched monograph. But it is a very enjoyable and thought-provoking read, and I think the mathematical community would benefit from more books like this one.

Darren Glass is a Professor of Mathematics at Gettysburg College. If he recalls correctly, his fantasy dinner party did include a mathematician, but also included Neil Gaiman, Bob Dylan, and several members of the 1991 Atlanta Braves. The list would only be slightly different today.

See the table of contents in the publisher's webpage.