8ECM: Interview with Karine Chemla

Questions:

1. What is your research domain?
2. Behind this diversity you describe, is mathematics all one?
3. How did your interest move from mathematics on to the history of mathematics and to the history of Chinese mathematics in particular?
4. Could you describe your interaction with other disciplines in your work as a historian of mathematics?
5. In short, you describe the way in which thought forges its tools for thinking?
6. What happened when you returned to France after your stay in China?
7. When did you join the CNRS and what changes did it bring to your career?
8. What is the history of the discipline "history of mathematics"?
9. Can you tell us about what you mean by "philosophy of mathematics"? How does this relate to your approach to the history of mathematics?
10. Tell us about research topics which have meant a lot to you?
11. What does it mean to you to be awarded the Otto Neugebauer Prize?
12. Is there a message you would like to pass on to the younger generations?

1. What is your research domain?

I am a historian of mathematics. At first glance, my research focuses on two quite different subjects: the history of mathematics in ancient and medieval China, and the history of mathematics in Europe in the first half of the 19th century. In fact, for me, these two subjects, and others to which I am devoted, are connected in a subterranean way, because what interests me in the history of mathematics are first and foremost theoretical questions. But in order to think about questions of this kind, it is essential to be able to dig into them in different historical contexts. We can thus evaluate how the questions that emerge when we work on a such domain are or are not relevant for such other one, or how those questions should be transformed in order to become relevant. The different theoretical questions I address are all aimed at understanding the same phenomenon, which is at the heart of my research: how come there are groups of people who do mathematics in different ways, and yet the mathematics they produce circulates and is eventually shared more widely.

That mathematics circulates and becomes shared seems to be an obvious fact, which we think we can explain by saying that it is "universal". Of course, 2 and 2 make 4 everywhere, but does that mean that everyone does mathematics in the same way and that everyone gives the same meaning to all statements? Certainly not. It is not the case if we look at a very ancient past, but it is not the case either if we look at mathematics today. Multiple human collectives share, at a given moment, a given way of doing mathematics; collectives which are never fixed for all eternity, collectives are formed, disappear, reform, divide or merge. If this were not the case, why would there be disagreements about the "right way" to do math? Why would there be communication difficulties between collectives?

I use the word "collective" here to avoid the word "community", which seems too rigid. I use it to grasp a fluid social reality that I think is important to characterize in order to understand mathematical activity, namely: that around common questions, groups of mathematicians regularly develop, who share enough ways of doing things to be able to work together. Today, these collectives are formed because people are connected by e-mail or by videoconference, because they meet during colloquiums or seminars, and because, by interacting, they share not only questions, techniques, and knowledge, but also figures and ways of using them, ways of writing and ways of demonstrating, etc., so that something collective builds up. At other times, it happens that members of a collective take an interest in questions, knowledge, or ways of doing things that have been developed and explored by another collective. Bridges are established, the two collectives will perhaps dissolve to form a new, larger one, unless another collective develops between the two former ones. A collective is, in my opinion, something quite fluid. It can be scattered all over the planet, or it can be created in an institution or in a place. It depends fundamentally on the means of communication available and the way mathematicians use them. Because for there to be a collective, things have to be able to flow between people in some way. I think that in the near future, we will have a history of collective action in mathematics, but this is just only the beginning.

It is the basic theoretical problem that interests me. And the fundamental reason I am interested in it is to answer the question "Why (and how) is there diversity in mathematics? ". There is a fairly common way of answering this question, which I consider deeply deviant: it consists in saying that "the Chinese" - sometimes it is because they speak Chinese, sometimes it is because they are Chinese - do mathematics differently from "the Europeans". This is a way of thinking that is extraordinarily widespread and, in my opinion, just as extraordinarily false. It challenges me that this representation is so widespread - it challenges me that something so obviously irrational is so widespread, but it challenges me above all because it amounts to a form of what I can only call: it is a form of racism. For it amounts to subscribing to the idea that there are human groups which, because they are a priori collectively different, and even, to force the point, different from all eternity, would do things differently, including mathematics.

The challenge, therefore, in my opinion, is not to deny that there is diversity - because there is diversity - but to understand its true nature at the end of a rational approach. It is also important to understand where diversity comes from, how it appears and how it disappears. In fact, if we look at the mathematical activity in China at a given time (which is what we did with the European SAW project), we find different collectives doing mathematics in different ways. And, as I said, I think it is the same if we consider a large mathematics department today: we will be able to see, if we look closely, that groups of mathematicians are doing mathematics in different ways. Of course, the key question is: what does "doing mathematics in different ways" mean? How can this be described? How do these ways emerge and disappear? These are precisely the questions that interest me.

2. Behind this diversity you describe, is mathematics all one?

That's the point. Take, for example, the so-called Pythagorean theorem. In the distant past, it could be seen as a statement that expressed a relationship between the lengths of the sides of a right-angled triangle, understood as geometric objects, and it could also be seen as a procedure that, from two numbers that expressed the lengths of two sides, allowed us to produce the third. Today, we move from one of these ways of seeing to the other in a very flexible way, so flexible that we no longer see the different statements that have been synthesized under the name of "Pythagorean theorem". But we find mathematicians of the past who have adopted the first way of seeing, others who have adopted the second way, and still others. Each approach has in fact given rise to different questions, all equally interesting. We see how the statement is both somehow the same and somehow different. The question is to know why it is different, what it refers to, but also how these two statements, as well as others, have been articulated to each other and made the same in the course of history. What did it take, what circulation of ideas, what transformations of ways of seeing the triangle, for these different statements to become points of view between which one could easily move? So my answer to the question: "Is mathematics all one?" is yes and no. Because in fact, what for us today is one and the same thing could have been understood in different ways in the past; and it could also appear tomorrow as the same thing as something that today seems to us to have no relation. It has been a long work in history for those things to be made, and the history of this work is just as interesting as the history of the introduction of a new concept or of the achievement of a new theorem.

3. How did your interest move from mathematics on to the history of mathematics and to the history of Chinese mathematics in particular?

We all know that we do what we do a little bit by chance... and I am no exception to this rather general rule. I was a student in mathematics at the École normale supérieure de jeunes filles. The director was Josiane Serre, a remarkable chemist and the wife of the mathematician Jean-Pierre Serre. Every two years, the Singer-Polignac Foundation offered travel grants to four students from the grandes écoles so that they could take a year's vacation, going far away to carry out a project they had formed. Josiane Serre imagined that I might be interested. I had just started a thesis on ergodic theory. She suggested I think about it. Two hours later, an idea came to me: "I'm going to do a project on science and culture. I still had to find a place where it would be particularly relevant to go and think about what was then only a vague idea. The answer came to me, and I said, "China". So it was a bit of a whim at first, but the Foundation gave me the grant. The funny thing is that I then had to obtain a visa - this was in 1980, not long after the Cultural Revolution - and when I called the Chinese embassy to ask for a visa to go to China and think about science and culture, they told me that at the time they did not grant visas for this kind of trip. So, in order to be able to go to China anyway, I invented that I wanted to go there to work on the history of mathematics in China. And... life took me at my word!

I left, and, if we could not wander about in China, I was, on the other hand, given the possibility to study there. Thanks to the mediation of Wu Wenjun, a Chinese mathematician who had studied in France and had met Jean-Pierre Serre there, the Institute of History of Natural Sciences, which is an institute of the Chinese Academy of Sciences - it is the equivalent of the CNRS -, created a program for me alone, and I financed the possibility of following it with my scholarship. They very generously gave me five researchers as professors to teach me the history of mathematics in China. However, these five researchers only spoke Chinese and Russian, but not French, English, or any of the languages I could have gibbered. Before I left, I had already understood what was in store for me, and so I had collected books to learn Chinese. I started to study the characters on my own and that's when I started to get really interested in this project. But when I arrived in Beijing by the Trans-Siberian Railway on April 6, 1981 at 3:30 p.m., I was not particularly well-versed. That said, after seven months of private lessons with teachers who spoke only Chinese and who taught me the history of mathematics in Chinese, I had made progress...

4. Could you describe your interaction with other disciplines in your work as a historian of mathematics?

Given my subject matter, I am in constant contact with sinologists since I have become a specialist in China, and more specifically I interact with colleagues who study all sorts of fields in China (linguistics, economic history, history of philosophy, and of course history of science). I also interact, of course, with mathematicians as well as with historians. Finally, I collaborate, because of some of my scientific interests, with linguists and anthropologists, because they bring me precious help to describe how the people I observe operate and how they work. So I find myself at the intersection of many disciplines. It is as exciting as it is demanding, because each discipline has its own criteria of rigor, different from those of other disciplines. I have to identify the criteria of rigor specific to each field with which I am in dialogue, and try to comply with all of these requirements, regardless of the person I am talking to. It is as difficult to talk about China to mathematicians or historians as it is to talk about history to linguists. You always have to adapt the way you talk to extremely different disciplines, but all these points of view are necessary to do the history of mathematics as I want to do it. Each of these disciplines offers questions, materials, and methods of approach that are useful for addressing the questions that I have.

As an illustration of the necessary diversity of approaches, I regularly hear that when one speaks a given language, this language would have an impact on the thought that one could develop by using it. Personally, I hold the opposite to be true. For me, language is something flexible, something that changes all the time, because speakers, by speaking, are working with it. In particular, a group of scientists evolves according to the problems they face and to work on them. This is obvious if we look, for example, at numbers. We can see that, in order to make calculations, researchers have introduced ways of writing that differ from the writing of the language and even from the enunciation of numbers. The traditional history of numbers has considered the notations of numbers - all the notations of numbers! - as deriving from the writing of the language. Historically, this is simply not true. The number notations with which we have calculated were not, as a rule, fashioned to write the words of the language, nor even from their writing. They are inscriptions that have been produced to work with. And this highlights the general fact that, when it comes to the language we work with, the language we write, we are constantly elaborating new ways of writing, new words, new ways of saying, new types of texts, new inscriptions, to work and to think. All this is very fluid. Language is not something so rigid that it determines how we think. And to better grasp these realities, cooperation with linguists is essential.

One of the things that I'm most interested in, one of the ways I approach the diversity of ways of doing science, is the kinds of texts that mathematicians work with. These types of texts obviously vary from one context to another. And one of my goals is to describe the invention of new types of texts and how these new types of texts are implemented, and can be correlated with the type of work that researchers do. To understand the relationship between the work one wants to do and the type of text one gives oneself to do that work, again, linguistics is essential to better address these questions.

5. In short, you describe the way in which thought forges its tools for thinking?

Absolutely. I take issue with the thesis that language constrains thought. As I said, for me, it is during the process of research that researchers form the languages with which they work, and the resulting languages that we must then describe are working languages, and not, as one would be tempted to think at first, "natural" languages, like French or Chinese. In my opinion, it is an illusion to imagine that one would work in a "natural" language. The languages in which we carry out research are artificial, and they are done, at each moment, in relation to what we do with these languages. I insist on the flexible and dynamic character of the languages we make, of the texts we make, of the inscriptions we make to work in mathematics. Thought gives itself tools in the form, in particular, of languages and texts, which it forges at the same time as it develops.

Instead of speaking of "the" mathematical language, we should in fact speak of a set of sub-languages, and we should look much more closely, in groups, at the languages that are really used and how they are used, in order to highlight this aspect of the work of the mathematician, which consists in elaborating his or her working tools. In my opinion, the language, the languages, are the products of mathematical activity as much as the concepts, the results, the theories. This is my thesis.

6. What happened when you returned to France after your stay in China?

I had been given a scholarship to spend a year on vacation and I hadn't really taken a vacation. I still had to write about a trip, so I came back and traveled through China. I spent a month traveling across the country, by train and by boat, and then I flew back.

At first I thought I was going to resume the mathematics thesis I had started before I left. But things took a different course. While I was still in China, I had started to write what I thought was an article about what I had studied there, and I had written, written and written... After a while, having written almost a hundred pages, without having finished, I said to myself, "I'm not writing an article, I'm writing something that is more substantial. Shortly after my return, I heard about Christian Houzel, who had heard about me from a mathematician I had met during my stay in China, Jean-Pierre Bourguignon, and at the first meeting Christian Houzel said to me: "If you want, I can make you do a thesis in mathematics with what you did in China. That's how I changed my subject - by chance, without really having premeditated it. I changed the subject by the logic of what I was doing.

7. When did you join the CNRS and what changes did it bring to your career?

When I had met Jean-Pierre Bourguignon in China, he had pointed out to me that the prospective report of the mathematics section of the CNRS emphasized the fact that it would be good to develop the history of mathematics within that section. When he told me this, it was September, five months into my stay in China, and it was before I started writing what was supposed to be an article and would become a river. I took note of the fact, but I didn't particularly incorporate it. On that I went home, and I found myself doing a thesis in mathematics with some history of mathematics. I applied to the CNRS on the basis of this work, and I had the immense good fortune that the mathematics commission, following the prospective report and on the basis of the first research I had already done, hired me to do the history of mathematics. This of course changed my life completely, since it allowed me to engage in this research what I had started out of pure personal interest without thinking that it would become a profession.

8. What is the history of the discipline "history of mathematics"?

The history of mathematics began to really develop, as a sub-discipline of mathematics, at the end of the 18th century, but it is especially in the 19th century that there began to be what signals the beginnings of professionalization, namely, specialized journals dedicated to the history of mathematics, and later, positions. It is also in the 19th century that a significant number of scholars devoted themselves to the history of mathematics, and even, for some of them, devoted the essential part of their research to it. Of course, one should nuance this. What I have said applies to Europe, and on the condition that we distinguish between those who were interested much earlier in the works of the past, but rather as the writings of colleagues, and those who, later on, adopted a more critical and more properly historical approach to ancient documents. On the other hand, the history of mathematics did not develop only in Europe, and in each region of the world, its development followed a different course and was motivated by other reasons. In fact, in the 19th century, whether in Europe, China or India, many mathematicians practiced philosophy or the history of mathematics as an integral part of their mathematical activity. This is much less the case today, and this is very regrettable, even if, of course, there are always exceptions, and fortunately so. It is therefore in the 19th century that the history of mathematics really began to become a discipline. However, at that time, it was practiced by all sorts of people. Besides mathematicians, there were, for example, civil servants of the East Indian Company, who resided in India, were interested in all sorts of things, including mathematical writings in Sanskrit, and who published on the subject. There were also Protestant missionaries who, working in China, studied, among other things, the mathematics of ancient China. We also find scientists who, by chance, learn Chinese, and will also touch upon the history of mathematics as well as many other subjects. These are therefore people who have received the most diverse training and are not all professionals in the sense that we understand it today, but who are beginning to publish articles in specialized or non-specialized journals, or even books.

If you take, for example, the Institute of Mathematics at the University of Göttingen in Germany, in the 1920s it was considered natural to develop, among other subjects, the history of mathematics, as well as the philosophy of mathematics. Otto Neugebauer, whose name was rightly chosen for the prize I am receiving, went to Göttingen to study mathematics and physics, and a few years later he was part of the team that published mathematician Felix Klein's course on the development of mathematics in the nineteenth century, he was writing a thesis on fractional calculus in ancient Egypt, and he was lecturing on the history of ancient mathematics. He developed the activity on the subject within the Institute of Mathematics to such an extent that it aroused the interest of many other mathematicians who were later to publish on the history of mathematics, notably Dirk Struik and Bartel van der Waerden. Unfortunately, the Göttingen Institute fell victim to the rise of Nazism in Germany, in particular due to the emigration of many of its most eminent members. Some of them who found themselves in New York were the founders of what is now the Courant Institute. Again, they implemented the same idea that the history and philosophy of mathematics should be an integral part of the mathematical activity. Morris Kline and Harold Edwards successively conducted research on these subjects.

Thus, on several occasions in history, mathematical institutes, and not the least, have actively considered that they should develop the history of mathematics among the subjects on which to conduct research. In fact, we can consider that this is finally the idea that the CNRS had made its own, just like, decades earlier, the Bourbaki group. Bourbaki's works systematically include historical notes, and I have always wondered whether this was not under the influence of the stays that its first members had made in Göttingen and in other German institutions.

9. Can you tell us about what you mean by "philosophy of mathematics"? How does this relate to your approach to the history of mathematics?

For reasons that probably have to do with my family background, I became passionate about mathematics quite early on, but my passion was increased tenfold by the readings on the philosophy of mathematics that I had been led to do in my senior year. My interest in the philosophy of mathematics is still alive today. Today, there are many ways to practice this activity. Here again, one can identify, in philosophy of mathematics, various groups, which follow different work programs. The main philosophical question that interests me is that of understanding the nature of the correlations that can be identified between the mathematical results that researchers produce and the specific way in which these researchers work. How does the way one conducts research become engraved in the results one produces, and conversely how do the questions one seeks to elucidate lead to the evolution of the way one does mathematics? What can we say about this relationship, and what does it tell us about mathematical activity? Today, many philosophers are interested in "mathematical practice", which means that the focus has diversified since the days when philosophers of mathematics focused primarily on the foundations of mathematics or on what was proven. Today, many of them also intend to look at mathematics as an activity and try to understand how this activity is practiced from a concrete point of view, how mathematicians work with diagrams or how they interact with the writings they create. These are questions that coincide with my own.

Indeed, my thesis is that, in order to identify mathematical collectives that interest me, one can rely on the way mathematicians make diagrams, on the way they elaborate their texts and inscriptions, on the way they calculate, etc. We then realize that the way in which diagrams are drawn has a history and presents variations from one collective to another. And the same is true for many other aspects of mathematical practice: the way in which mathematical activity is carried out has a history, and the reason for this is, in my opinion, that mathematicians develop ways of making diagrams, inscriptions, ways of writing, etc., at the same time as they develop demonstrations and establish results. This is the reason why their mathematical practice is constantly changing and why the collectives can be understood from these distinct ways of doing mathematics. What interests me, to put it in other words, is, on the one hand, this joint process of making the tools of work and producing results or theories, and, on the other hand, how things that have been made locally circulate: how they are taken up by others, how they are thus transformed, and how they become shared things. Generally speaking, we always have tools that are similar: in general, mathematicians draw and calculate. However, if we look closely, we can also see notable differences in the way figures and calculations are implemented. It is always, although on another level, the play of the same and the different, which can be observed on a mathematical result, as well as on the use of diagrams, the writing of numbers or the conduct of calculations.

10. Tell us about research topics which have meant a lot to you?

Several topics have been particularly crucial in my research. The history of mathematical demonstration has been one of them. I became interested in this subject on the basis of Chinese texts dating from the third century A.D. that set out mathematical demonstrations. However, these demonstration texts are not the same as those found, for example, in the Elements that a certain Euclid finished writing in Greek around 300 BC. However, until a few decades ago, the history of demonstration could be reduced to this: "Demonstration was born in Greece" and "we", because we attach importance to demonstration, axioms and definitions, "are in the line of the Greeks" - that is indeed how they spoke at the time. Colleagues who began to work on Chinese texts of the type I mentioned earlier came up against the same difficulty: for them, as for most people, demonstration means Euclid, and so they first tried to show that the Chinese texts in question contained demonstrations comparable to those of Euclid. But this analysis was far from satisfactory, simply because these demonstrations did not fit into this mold and one had to tamper with the texts to claim the contrary. In reality, the method with which these texts were approached was not satisfactory. In fact, one started from an a priori idea of what a demonstration should be and, basically, the question one asked was to know whether or not the ancient Chinese texts were in conformity with this idea. By proceeding in this way, one writes the history of what one thinks should be a demonstration, but not the history of what various groups did when they tried to establish that a statement was true or that a calculation procedure was correct. Now, the critical problem with the a priori idea was that it was particularly rigid and did not correspond to what mathematicians do when they demonstrate. To apply it systematically would have been to erase from the history of demonstration René Descartes, Leonhard Euler, Jean-Victor Poncelet, Henri Poincaré and many others, which is something I personally do not want to do. I therefore became interested in Chinese demonstration texts in order to reflect on what it meant to "demonstrate" in mathematics. That is to say, instead of assuming that I knew what "demonstration" meant and that I knew the motivations of mathematicians to engage in this exercise, I instead asked myself what these Chinese texts taught me about demonstration. In particular, is the purpose of demonstrating really limited to establishing that something is true, as has been widely said, or is it an activity with much more complex ins and outs? Since philosophers of mathematics have started to focus on the real practice of mathematics, and not on an idealized practice, we have started to realize the complexity of the activity of demonstration and the multiplicity of its aims. In this context, it becomes clear that reducing the exercise to the sole objective of establishing that something is true and judging the validity of an argument by its own criteria clearly represents an impoverishment for historical inquiry. For my part, I have therefore undertaken a joint reflection on what it means to demonstrate, and an attempt to situate these Chinese texts in a consequently broader vision of the history of demonstration. I conducted this research in the context of a collective project in which colleagues working on Greece, Mesopotamia, and India participated. We also dug into the history of the history of mathematical demonstration to understand why the ideas about the history of mathematical demonstration that I mentioned earlier were so widespread and so persistent. Where did they come from, how did they take shape? This has been one of the big projects of my life.

Another important task for me was to understand the different ways of doing mathematics. It has been to try to understand what it means that the ancient Chinese texts that have come down to us are composed of problems and procedures that solve these problems. These texts have often been approached as if they were collections of exercises or reference books for civil servants who needed to find recipes for solving the problems they faced in the context of their administrative work. Once again, it was a preconception, this time based on our experience with comparable texts today. But, again, it became clear to me that we needed to find another approach, a truly historical one, to determine how, in ancient China, readers of this type of text approached them, how they interpreted them. And one of the key questions in this context was to ask what was a problem for them. Clearly, this question addresses one of the central elements of the way of doing and writing mathematics that these texts attest to. In these contexts, practitioners had typically put problems and procedures at the heart of their mathematical activity. So learning to read them, by finding documents that shed light on how ancient Chinese practitioners used them, meant acquiring key tools for interpreting these writings. Now, as soon as we ask ourselves this question, we can find many clues, which show that a problem was not, in these contexts, simply a question to be solved and that the practitioners of mathematics had a theoretical reflection on the procedures attached to these problems. The crucial question from which my approach to mathematical problems proceeds can be generalized to any practice of mathematics: can we identify documents that allow us to describe this practice in a contextualized way? This is where anthropology comes in, in order to avoid the pitfall of anachronism, which is the first sin of a historian and which amounts to assuming that the people of the past that we observe used mathematical problems as we do, to take this example.

11. What does it mean to you to be awarded the Otto Neugebauer Prize?

What it means... First of all I have to say that receiving an award named after "Otto Neugebauer" emotionally means a lot to me. Otto Neugebauer, who had only moral reasons to leave Nazi Germany - he had none of the most common reasons to be persecuted - slammed the door and refused to obey any of the outrageous laws he was asked to implement. He is, in particular, a man who chose to resign from the editorial board of the newspaper he founded rather than obey the order to exclude his Jewish colleagues. He embodies values to which I adhere, values that are rare and difficult to implement. He is a high moral authority, hence the emotion I feel, I must emphasize, in receiving a prize that bears the name of this researcher whom I respect infinitely.

All those who conduct research on ancient mathematics have had to study the publications of Otto Neugebauer. One has no choice, he was really a central author. But I also worked on him a little bit as a historian, and that's how I came to understand who he was. I can be happy that it was his name that was chosen for this European prize.

That's the first thing.

And then, in the research that I carried out, I never took the highways. I did not take the same paths as most of my colleagues, working on a subject that could be thought of as marginal in every respect with respect to the discipline of the history of science. I did it in particular with the idea that it could lead to theoretical renewals. And it is a great pleasure to be able to say to young people who want to turn to research that it is not by following fashion, it is not necessarily by doing what everyone else is doing, that one does research. One does research with convictions, by having one's own questions, by having one's own agenda, by listening to one's desire to answer certain questions, and by making means for that. To be able to give this advice to younger people is something that is important to me. I think that the way research funding is allocated today leads to sacrificing too much to the same subjects and that there is not enough support for people who do other things. I would like to see a conviction that there are important things to be gained by giving everyone the opportunity to pursue each other's questions rather than the latest fad that is more likely to secure you funding today.

One value that is important to me, especially in research, is authenticity. Being authentic means that the questions you work on come from you, from what you need to understand, not from where you get funding. In my opinion, if you don't practice research in this way, it can't be a lasting research. That is my firm belief.

14. Is there a message you would like to pass on to the younger generations?

Surely, a twofold message. The first message comes from Otto Neugebauer, who in the name of moral principles refused to accept unworthy laws and preferred to leave everything, who even refused to speak German in the second half of his life. I particularly wish that young people have the possibility to follow moral principles in their research and that they are not put in a position where they have to compromise with ethics and morals in order to succeed.