5.2.1. A conception of innovation, inherited from the conception of Henri Poincaré’s mathematical invention

Cédric Villani begins many of his lectures with a reference to Henri Poincaré, who he says is “one of the few to have tried to answer the question: where do ideas come from?” [MAR 12]. Indeed, Henri Poincaré was one of the first mathematicians to expose, in his famous lecture entitled L’invention mathématique in 1908, his reflections on the process of mathematical invention, based on his own experiences. He describes a process in four key stages, mixing “conscious” and “unconscious”²work phases. In the following, we briefly describe these four stages and show how Villani draws inspiration from them to describe the genesis of his theorem.

Henri Poincaré calls the first stage the “preparation or the initial conscious work”. According to him, the study of an issue begins with the mobilization of ideas in order to create links between them in order to build the desired solution. This phase of work alternates progress, failure and recovery. While this phase of the work is uncertain regarding the production of results, it provides a good understanding of the problem and its difficulties. In his book, Cédric Villani mentions this first stage of the mathematical invention process in these terms: “a mathematician’s first steps into unknown territory constitute the first phase of a familiar cycle” [VIL 15, p. 243]. He details this black period when he begins his research on Landau damping: a first phase of work devoted to determining the question “but in order to solve a problem, you’ve got to know at the outset exactly what the problem is! In mathematical research, clearly identifying what it is you are trying to do is a crucial, and often very tricky, first step” [VIL 15, p. 12]. Then, he takes “I have succeeded in clearing a major hurdle: now I know what I want to prove” [VIL 15, p. 32]. Then follows a second phase of work: possessed by the problem “and the days and the nights passed in the company of the problem” [VIL 15, p. 37], he explores “paths and subpaths, meticulously noting every possibility, crossing off dead ends” [VIL 15, p. 37]. There are small advances but “No, the problem hasn’t been cracked yet” [VIL 15, p. 38]. We find the key elements described by Henri Poincaré in this phase of work: delimitation of the problem, alternating progress, failures and new ideas.

The second stage of the mathematical invention process described by Henri Poincaré is “incubation or the role of unconscious work”. Where efforts seem unsuccessful after a period of conscious work, a period of abandonment of the problem frequently follows. Poincaré described a moment when he thought of something else: “Disgusted with my failure, I went to spend a few days by the sea, and I thought of everything else” [POI 93, p. 145]. This stage is characterized by an unconscious work whose role is to exercise a certain separation of the various ideas and established results and to make available only those which are useful to the conscious mind. Cédric Villani refers to this stage in these terms: “During the time it took to get here the story had flowed through my brain and trough my veins, a small torrent of ink and paper. I felt cleansed through and through. While I’m reading manga all thoughts of mathematics are suspended […] the question echoed through my mind over and over again. If there really is a connection, I’ll find it” [VIL 15, p. 18]. However, this phase of incubation and unconscious work is necessary but not sufficient: illumination must still arise.

“Illumination” is the third key stage in the process described by Henri Poincaré. It is the consequence of unconscious work. Henri Poincaré describes this stage as the following criteria: brevity, suddenness and immediate certitude. Indeed, illumination does not last and it is unforeseen. It comes unexpectedly, sometimes when busy with other tasks, as evidenced by this famous quotation by Henri Poincaré, “As I was stepping on the step, the idea came to me, without anything in my previous thoughts seeming to have prepared me for it” [POI 93, p. 145]. As for the immediate certitude, it has a strong impression but is not a demonstration.

Cédric Villani also refers to these two stages when he describes the genesis of his theorem. He explains that after sometimes unsuccessful efforts, it is new ideas or tactics that allow him to take up his work again. Then, one of them will be decisive: “After the darkness comes a faint, faint glimmer of light, just enough to make you think that something is there, almost within reach, waiting to be discovered…” [VIL 15, p. 243]. In the interview with the newspaper Le Nouvel Observateur [GRU 12], he explains that “illumination can occur at the most fortuitous moment, when one was busy with something else, and thoughts follow one another as if by miracle”. In his book, he recounts this particular moment in his research on Landau damping:

I hear a voice in my head. You’ve got to bring over the second term from the other side, take the Fourier transform and invert to L². Unbelievable! […] I rush back home and settle down in an armchair to try out the idea that came to me when I woke up, as if by magic. […] I go on scribbling, then pause for a moment to reflect. It’s works! I think… YES! It works!!! Of course […] [VIL 15, pp. 141–142].

This voice appeared to him suddenly and briefly, upon waking up and a few hours later, he is sure that this idea will provide him with a key for the continuation of his work. These remarks are very close to those used by Henri Poincaré, notably the ideas of immediacy, brevity and certitude.

Finally, the fourth and last stage proposed by Henri Poincaré is the “second period of conscious work”. The first step is to verify the very idea of illumination. Even if there is a sense of certitude this idea may be wrong. This requires the intervention of reason in order to verify the result, to write it down precisely, and then to draw consequences and possible applications. Cédric Villani’s account clearly refers to this stage: “Then after the faint, faint glimmer, if all goes well, we unravel the thread, and suddenly it’s broad daylight! You’re full of confidence, you want to tell anyone who will listen about what you’ve found” [VIL 15, p. 243]. This moment marks the “end of open-ended exploration” and the beginning of a consolidation and verification of the work: “Now we’ve got to consolidate, reinforce, verify, verify, verify… the moment has come for us to deploy the full firepower of our analytical skills!” [VIL 15, p. 110]. This phase of research thus marks the return to conscious work, accompanied by a change in work method “the effort of imagination to find a logical sequence” gives way to “an effort of rigor so that it is good” [JAM 10]. These remarks echo those of Henri Poincaré who stressed this when he said that “it is by logic that we prove, it is by intuition that we invent” [POI 24, p. 137].

From the above, it appears that the process of innovation genesis described by Cédric Villani is largely inherited from the conception of the mathematical invention of Henri Poincaré. But Cédric Villani goes further than Henri Poincaré, as we will see in the following. Other factors play a key role, which leads him to identify what he calls the seven key ingredients of “innovation ideas”. Let us break down these ingredients.

5.2.2. The seven ingredients of “innovation ideas” according to Cédric Villani

The first key ingredient of innovation according to Cédric Villani is “documentation”. As he says in his lectures, you have to “feed the brain” for an idea to emerge. He mentions various sources of documentation: encyclopedias, experimental results, information, representations of results, formulae, theorems, etc. He specifies that these documents, put in memory and then remobilized at the time of the emergence of the idea, come to “feed the reflection” and that it is “from there one can have ideas of what is possible, of what will make sense […]” [MAR 12]. He illustrates this role of documentation by taking an example in the genesis of his theorem:

In the genesis of the theorem, I convinced myself that I needed […] a certain formula […] I remembered that I had seen it 15 years before […] but I had no idea of the name of the formula itself but with a search engine, in a few seconds, I could find the name of the formula, the formula, and I had all the history [MAR 12].

The second key ingredient is “motivation”, which Villani says is “the most important and mysterious ingredient of all. Nobody really knows what motivates people” [EPI 16]. In his book, the genesis of his theorem is presented as an adventure tale full of pitfalls, made of incessant comings and goings, of failures and moments of doubt. He explains the role of motivation in overcoming all these obstacles: “We believe in it and it is irrational” [EPI 16] and even “Nonetheless, I remain convinced that nothing can stop us. My heart will conquer without striking a blow” [VIL 15, p. 100]. To encourage motivation, he mentions the awakening of curiosity through different media.

The third key factor is, according to Cédric Villani, “a favorable environment”, both material and human conditions. He evokes the importance of creating an ecosystem to interact with other people: “you need to develop the idea, an environment in interaction with other human beings” [MAR 12]. According to him, the École Normale Supérieure (ENS) de Lyon (a graduate school in Lyon) played a central role in the genesis of his theorem, notably the discussions between researchers from different fields: “That’s what I love most of all about our small but very productive laboratory – the way conversation moves from one topic to another, especially when you’re talking with someone whose mathematical interests are different from yours. With no disciplinary barriers to get in the way, there are so many new paths to explore!” [VIL 15, p. 17]. The fact of having discussions between disciplines led him to make links between geometry, analysis and probabilities. This is also what he appreciates at the Institut Poincaré [Poincaré Institute], which welcomes researchers from all over the world and whose “mission is to provide these researchers with the conditions so that they can brainstorm together, interact together, develop their ideas, have new ones and develop those” [MAR 12]. He also states that this enabling environment can be citywide.

The fourth ingredient recognizes the “collective dimension of innovation”. It is about the importance of relationships. He specifies that if face-to-face exchanges are fundamental for the beginning of a project, for the “creation of ideas”, they can be held remotely by e-mail, to shape and develop ideas. He explains that in February 2009, he “exchanged a good hundred e-mails with Clément [Mouhot]; in March, more than two hundred!” [VIL 15, p. 111]. The exchanges allow progress to be made, thanks to the confrontation of ideas and perspectives of the problem: “a new thinking entity [which] emerges, it is not him, it is not me, it is the combination of the two and the game between the two that makes […] a sort of new way of thinking, like a delocalized brain” [MAR 12].

The fifth ingredient relates to “constraints”: “If you are not constrained by the right factors, you will not be motivated to have new ideas” [EPI 16]. He explains that constraints “force us to explore things in more detail”, to have more imagination [MAR 12]. To illustrate his point, he drew a parallel with the arts and cited two examples: a piece of music, composed solely of the note “A” by György Ligeti, and The Disappearance, a 300-page novel by the writer Georges Perec written with the feat of not using the letter “e”. In both cases, the authors were forced to use other melodies or phrases to create their work. At the level of mathematics, he also evokes this role of constraints and explains, for example, that he sets for himself a constraint to solve mathematical problems, that of proposing demonstrations only with constructive arguments³.

The sixth ingredient is “work and intuition”. The genesis of his theorem lasted more than 2 years. Indeed, good ideas do not come alone but are the result of much trial and error. His journey also reveals that ideas often arise from alternating phases of intense reflection and sudden illumination as he explains in one of his lectures:

In my work, I give many external elements […] that will influence on the little spark that you did not expect, that does not exempt you from working hard, because it is usually in this changeover between the phase of hard work where you soak up the problem and the phase of illumination that critical solutions occur in [MAR 12].

He then agreed with Henri Poincaré and endorsed Louis Pasteur’s thesis that “chance favors invention only for minds prepared for discovery by patient study and persevering effort” [PAS 29, p. 132].

The seventh and final ingredient is “perseverance and luck”. There are many traces of Cédric Villani’s perseverance in his book: “If there really is a connection, I’ll find it. At the time I had no way of knowing that it would take me more than a year to find the link between the two” [VIL 15, p. 18] and “[…] Then, suddenly, a new hole opens up. Hopping mad, I’ve just about had it. Had it up to here with this whole business!” [VIL 15, p. 139]. He also specifies that most of the time new ideas don’t work and so “if you can’t stand to be perplexed, you shouldn’t do the mathematician’s job” [EPI 16]. He also mentions multiple moments when luck has smiled on him: “From time to time, a stroke of luck happens. You have to spot it and jump on it” [EPI 16], using many that sound religious and divine: “I went into semi-automatic pilot, drawing on the whole of my accumulated experience…but in order to be able to do this, first you’ve got to tap into a certain line – the famous direct line, the one that connects you to God, or at least the god of mathematics. Suddenly, you hear a voice echoing in your head. It’s not the sort of thing that happens every day, I grant you” [VIL 15, p. 141], and “Second miraculous coincidence had occurred – and on the same subject!” [VIL 15, p. 18], and finally “and then the miracle occurred. Everything seemed to fit together as if by magic” [VIL 15, p. 126]⁴.

We can only subscribe to the seven ingredients identified by Cédric Villani, which have been highlighted on numerous occasions. The importance of documentation in innovation is indeed a widely accepted idea. It is based on the observation that innovation does not emerge in a desert but presupposes a knowledge base already there to innovate [PER 01]. It also goes hand in hand with the idea of incremental mathematics defended by Cédric Villani, according to which progress is made mainly by improving the work of others [VIL 15, p. 123]. The question of the assimilation of documentation refers to the rich literature dedicated to innovation under the term “absorption capacity” in the wake of the work of Wesley Cohen and Daniel Levinthal [COH 90]. Similarly, the emphasis on motivation updates Teresa Amabile’s [AMA 83] idea of “flow” in psychology or “intrinsic motivation”. The collective dimension joins the works that challenge the myth of solitary genius. For example, Albert Einstein, who for a long time was presented as a solitary genius, was in fact perfectly aware of what was happening in the university world and in particular of the work of Henri Poincaré and Edward Lorentz [BAL 08]. The emphasis on the importance of exchanges thus echoes the research that emphasizes that the innovator is not an isolated actor and that his or her capacity for creation results from his or her relationship to others [ISA 15]. It also aligns with the work of Carl Rogers [ROG 05], who emphasizes that communication allows his results to be compared with the opinions of other experts. If work, intuition, perseverance and illumination refer to Henri Poincaré’s conception, these are again factors commonly accepted as vehicles of creativity and innovation, as the study conducted by Mihaly Csikszentmihalyi [CSI 06] of 91 great inventors and creators (1996) attests.

5.3.1. A network of actors with varied knowledge

When reading Théorème vivant, we are immediately struck by the spectrum of actors with whom Cédric Villani interacted during the conception of the theorem. Among these actors, Clément Mouhot, his former doctoral student, occupied a central place. Throughout the book, he presents the interactions they had that established a new result. It all began with a working session on March 23, 2008, when Cédric Villani wanted to talk to Clément Mouhot about a problem. This was followed by a mathematical discussion in which each provided answers to the other’s questions and, above all, asked new questions:

I paused to explain why, at some length. We discussed. We argued. […] Clément was still unsure about the positivity. […] Each of us set about reconstructing the argument that this postdoc, Dong Li, must have developed. […] We scribbled away in silence for a few minutes. I won. […] Clément was thinking out loud, staring at my calculation. Suddenly his face lit up. In a state of great excitement, he jabbed at the board with his index finger: But then you’d have to check to see whether that helps with Landau damping! I was at a loss for words. […] Now it was my turn to ask Clément to explain. He didn’t know what to say either. He hemmed and hawed, shifting his weight from on foot to the other. Then he said that my solution reminded him a conversation he’d had three years ago with a Chinese-born mathematician […] [VIL 15, pp. 5–7].

This working session was the starting point of a close collaboration that would last several months, a collaboration that essentially took the form of correspondence supported by e-mails⁵ due to working several thousand kilometers from each other.

In addition to this privileged collaboration with Clément Mouhot, Cédric Villani met many other actors with diverse knowledge. Without pretending to be exhaustive, we can cite his meeting with Freddy Bouchet on April 2, 2008. The latter, who has a dual background as a mathematician and physicist, is interested in the spontaneous organization of stars in a stable configuration. At the end of this meeting at the ENS of Lyon, his office neighbor, the mathematician Etienne Ghys, member of the French academy of sciences, discussed with him the Kolmogorov–Arnold–Moser theorem (KAM theorem). On January 29, 2009, he went to the seminar of statistical physics at the University of Rutgers accompanied by Eric Carlen, specialist in gas theory, and Joel Lebowitz “the high priest of statistical physics” [VIL 15, p. 119]. That same day he met Michael Kiessling, a plasma physics specialist. “Now, at the table, Michael was telling me with his usual infectious enthusiasm about how as a young man he fell in love with plasma physics, screening, the plasma wave echo, quasi-linear theory, and so on” [VIL 15, p. 87]. He collaborated with Francis Filbet, a colleague from Lyon who provided him with images of the Landau damping on March 11, 2009. On March 24, 2009, he met Elliot Lieb, one of the most famous and feared mathematical physicists at Princeton, and on October 27, 2009, J. Rauch, the specialist in partial differential equations, at Ann Arbor (University of Michigan). Figure 5.1 summarizes and illustrates this network of actors with whom Cédric Villani interacted throughout his research, during informal meetings (e.g. during a meal, a coffee break or discussion in an office) or more institutional meetings (notably research seminars).

5.3.2. Contribution of the network of actors to the genesis of the theorem

If the preceding meetings attest to the collective dimension of the mathematical research process, we also clearly perceive their contribution to the genesis of the theorem. From his meeting with Freddy Bouchet, Cédric Villani said “it was a profitable conversation” [VIL 15, p. 15], from his collaboration with Clément Mouhot he draws the following assessment:

In the end, Clément and I will be able to share the credit for the major innovations of our work more or less equally: I came up with the norms, the deflection estimates, the decay in large time, and the echoes, he came up with the time cheating, the stratification of errors, the dual time estimates, and now the idea of dispensing with regularization. And then there’s the idea of gliding norms, a product of one of our joint working sessions; not really sure whose idea that was [VIL 15, pp. 109–110].

But these encounters also make it possible to become aware that the genesis of the theorem is made possible thanks to the resonance of knowledge that belongs to distinct universes. Cédric Villani and Clément Mouhot used knowledge from static physics, mathematical analysis (differential calculus), statistics and geometry. His meeting with Michael Kiessling also gives him a glimpse of a possible analogy:

His mention of the plasma echo immediately concentrated my full attention. […] All this brought to mind some calculations I did a few days ago: a temporal resonance…the plasma reacting at certain quite specific moments…I thought that I’d lost my mind – but perhaps it’s the same thing as the echo phenomenon that plasma physicists have known about for years? [VIL 15, pp. 86–87]

His account of the genesis of his theorem also allows us to see how the criticisms addressed by his peers Elliot Lieb and Greg Hammet during his first seminar in Princeton on March 24, 2009 allow him to see his problem under a new light:

During the past week I’ve learned so much from lecturing on Landau damping. After my first talk, once his irritation had subsided, Elliott shared some valuable insights into the conceptual difficulties of the periodic Coulomb model. In the second talk, I laid out the main physical ideas of the proof. Elliot very much appreciated the mixture of mathematics and physics […] [VIL 15, p. 134].

When reading his book, we also perceive how his criticisms allowed him to improve his results. This is precisely what happened at the end of his meeting with Jeff Rauch. Troubled by a criticism of the latter, he laid out on paper a reasoning intended to convince the latter and suddenly “It is the illumination […] the fate of the item has just changed” [VIL 15, p. 217].

As Cédric Villani points out “each time he is at a meeting that triggers everything” [VIL 15, p. 147]. Indeed, the meetings allowed him to establish new relationships, thereby opening the field of possibilities. In doing so, he used his creative rationality to advance the conception of his theorem.

5.4.1. Creative rationality: what are we talking about?

Creative rationality is the ability to bring apparently distinct worlds closer together, to find links where they did not exist, to see what is happening outside one’s profession and to be open to everything. It is thought based on relationship:

As Poincaré (1913) noted “To create consists of making new combinations of associative elements which are useful” (p. 286). Creative ideas, he further remarked, “reveal to us unsuspected kinships between other facts well known but wrongly believed to be strangers to one another” (p. 115) [MAR 05, p. 137].

It is thus the source of innovation [JAC 94]. It is similar to the idea of bisociation as defined by Koestler: “it uncovers, selects, reshuffles, combines, synthesizes already existing facts, ideas, faculties, (and) skills” [KOE 64, p. 120] by putting forward the encounter of elements belonging to two distant universes by “thinking aside” and which, thanks to “clandestine games of the mind” [KOE 64, p. 31], opens on a creative synthesis.

This knowledge crossing is an adventurous transgression [FOR 18]. To cross knowledge, to combine knowledge that belongs to distinct universes leads indeed to detach oneself from established norms and paradigms. This is echoed by Arthur Koestler who, in his book The Act of Creation, pointed out that when we are faced with a problem we have already faced, we solve it with solutions that have already proved their worth. These solutions become routines that must be broken to be creative: “The act of discovery has a disruptive and a constructive aspect. It must disrupt rigid patterns of mental organization to achieve the new synthesis” [KOE 64, p. 104]. This is the intuition that Cédric Villani very quickly developed when he conceived his theorem “deep down I am convinced that the solution will require completely new tools, which will allow us to look at the problem in a new way” [VIL 15, p. 38].

Moreover, if we say that this transgression is adventurous, it is because it contains all the ingredients of an adventure. Deploying one’s creative rationality leads one to explore lands where no one has dared to venture before because, to use the words of Gilles Garrel and Elmar Mock: “We innovate by taking crossroads, where others are not, because we do not pick mushrooms on highways but on small paths” [GAR 12, p. 1]. However, when one follows such paths, one quickly faces many adventures, which require not only a solid motivation, but also perseverance to face them. Cédric Villani points out in the genesis of his theorem that he “never thought it would be so hard. No way I could have foreseen the obstacles that lay ahead!” [VIL 15, p. 74].

5.4.2. Cédric Villani and creative rationality

The genesis of Cédric Villani’s theorem is a perfect illustration of the deployment of creative rationality. From the very first pages, we became aware of the role played by Clément Mouhot – “Today I’m the one who needs help. The problem I’ve chosen to work on is exceedingly difficult” [VIL 15, p. 5] – with whom he will exchange abundantly during all of the conception process of the theorem. But, as we have seen before, the genesis of the theorem also owes much to the meetings and discussions that dot the course of this adventure. These meetings give him access to other points of view and knowledge which, as we have seen previously, belong to various fields. The deployment of creative rationality also implies detachment from the established norms “I need a new norm” [VIL 15, p. 38], openness to the unexpected: “the appearance of Faà di Bruno’s formula is symptomatic of the unexpected combinatorial turn that our work takes” [VIL 15, p. 60].

The genesis of his theorem is born of an ingenious combination of knowledge that Cédric Villani evokes via the metaphor of marriage – “now we have understood that we must marry the two points of view” [VIL 15, p. 117] – or by referring to the idea of resonance: “In my own research, the most astonishing comes from the resonance of apparently unrelated subjects. This sometimes results in astonishing ricochets […] which, adjusted to the rest, miraculously gives us the solution to a puzzle on which we have been stumbling for months” [Villani in EER 14]. Cédric Villani’s “genius” lies thus in his ability to cross the boundaries of knowledge mapping and synthesize knowledge – “In addition to the joint influence of these four teachers, I incorporated other elements and create my own mathematical style” [VIL 15, p. 134] – and create links where there were none: “It was as though I had acted as a catalyst!” [VIL 15, p. 136]. It was the hidden links between different mathematical fields that made his reputation as a researcher. “These connections are invaluable! It’s a bit like a game of ping-pong: every discovery you make on one side helps you discover something new on the other” [VIL 15, p. 134]. This knowledge crossing can be described as similar to an adventurous transgression “from thread to thread, one travels through mathematical questions” [VIL 15, p. 14]. As with any adventure, it has nothing of a long quiet river: “we took notes, we argued, we got annoyed with each other, we reached agreement about a few things, we prepare a plan of attack” [VIL 15, p. 7], “and days and nights passed in company with the problem” [VIL 15, p. 37]. The adventure was full of obstacles that our “hero” will overcome. “This latest difficulty does indeed look formidable, I admit. But still I feel sure we will find a solution. Already three times in the last three weeks we’ve found ourselves at an impasse, and each time we’ve found a way around it” [VIL 15, p. 100].