France has an outsized influence in the world of mathematics despite having significantly fewer resources than countries like the United States. With approximately 1/6th of the US population and 1/10th of its GDP, and French being less widely spoken than English, France’s mathematical achievements are remarkable.
This dominance might surprise those outside the field. Looking at prestigious recognitions, France has won 13 Fields Medals compared to the United States’ 15 a nearly equal achievement despite the vast difference in population and resources. Other European nations lag significantly behind, with the UK having 8, Russia/Soviet Union 6⁄9, and Germany 2.
France’s mathematicians are similarly overrepresented in other mathematics prizes and honors, confirming this is not merely a statistical anomaly.
I believe two key factors explain France’s exceptional performance in mathematics while remaining relatively average in other scientific disciplines:
1. The “Classes Préparatoires” and “Grandes Écoles” System
The French educational system differs significantly from others through its unique “classes préparatoires” (preparatory classes) and “grandes écoles” (elite higher education institutions).
After completing high school, talented students enter these intensive two-year preparatory programs before applying to the grandes écoles. Selection is rigorously meritocratic, based on performance in centralized competitive examinations (concours). This system effectively postpones specialization until age 20 rather than 18, allowing for deeper mathematical development during a critical cognitive period.
The École Normale Supérieure (ENS) stands out as the most prestigious institution for mathematics in France. An overwhelming majority of France’s top mathematicians—including most Fields Medalists—are alumni of the ENS. The school provides an ideal environment for mathematical talent to flourish with small class sizes, close mentorship from leading mathematicians, and a culture that prizes abstract thinking.
This contrasts with other countries’ approaches:
Germany traditionally lacked elite-level mathematical training institutions (though the University of Bonn has recently emerged as a center of excellence)
The United States focuses on mathematics competitions for students under 18, but these competitions often emphasize problem-solving skills that differ significantly from those required in mathematical research
The intellectual maturation between ages 18 and 20 is profound, and the French system capitalizes on this critical developmental window.
2. Career Stability Through France’s Academic System
France offers significantly more stable academic positions than many other countries. Teaching positions throughout the French system, while modestly compensated, effectively provide tenure and job security.
This stability creates an environment where mathematicians can focus on deep, long-term research without the publish-or-perish pressure common in other academic systems. In mathematics particularly, where breakthroughs often require years of concentrated thought on difficult problems, this freedom to think without immediate productivity demands is invaluable.
While this approach might be less effective in experimental sciences requiring substantial resources and team management, for mathematics—where the primary resource is time for thought—it has proven remarkably successful.
I don’t buy your factors (1) or (2). Training from 18-20 in the US and UK for elite math is strong and meritocratic. And brilliant mathematicians have career stability in the US and UK.
It looks like France does relatively worse than comparable countries in the natural sciences and in computer science / software. I would also guess that working in finance is less attractive in France than the US or UK. So one possible factor is opportunity cost.
The UK/US system typically gives tenure around ~40, typically after ~two postdocs and a assistant → associate → full prof.
In the French system a typical case might land an effectively tenured job at 30. Since 30-40 is a decade of peak creativity for scientists in general, mathematicians in particular I would say this is highly
Laurent Lafforgue is a good example. Iirc he published almost nothing for seven years after his PhD until the work that he did for the Fields medal. He wouldnt have gotten a job in the American system.
He is an extreme example but generically having many more effectively tenured positions at a younger age means that mathematicians feel the freedom to doggedly pursue important, but perhaps obscure-at-present, research bets.
My point is primarily that the selection is at 20, instead of at 18. It s not about training per se, although here too the French system has an advantage. Paris has ~ 14 universities, a number of grand ecolees, research labs, etc a large fraction which do serious research mathematics. Paris consequently has the largest and most diverse assortiment of advanced coursework in the world. I don’t believe there is any place in the US that compares [I’ve researched this in detail in the past].
The UK does not have the same tenure system as the US. I believe top mathematicians have historically (i.e. last 70 years) often become permanent lecturers fairly young (e.g. by age 32).
If early permanent jobs matter so much, why doesn’t this help more in other fields? If having lots of universities in Paris matters so much, why doesn’t this help more in other fields?
I wouldn’t claim to be an expert on the UK system but from talking with colleagues at UCL it seems to be the case that French positions are more secure and given out earlier [and this was possibly a bigger difference in the past]. I am not entirely sure about the number 32. Anecdotally, I would say many of the best people I know did not obtain tenure this early. This is something that may also vary by field—some fields are more popular, better funded because of [perceived] practical applications.
Mathematiscs is very different from other fields. For instance: it is more long-tailed, benefits from ′ deep research, deep ideas’ far more than other fields, is difficult to paralellize, has ultimate ground truth [proofs], and in large fraction of subfields [e.g. algebraic geometry, homotopy theory …] the amount of prerequisite knowledge is very large,[1] has many specialized subdisciplines , there are no empirical
All these factors suggest that the main relevant factor of production is how many positions that allow intellectuall freedom, are secure, at a young age plus how they are occupied by talented people is.
e.g. it often surprises outsiders that in certian subdisciplines of mathematics even very good PhD students will often struggle reading papers at the research frontier—even after four years of specialized study.
I’d defend a version of claim (1): My understanding is that to a greater extent than anywhere else, top French students wanting to concentrate in STEM subjects must take rigorous math coursework from 18-20. In my one year experience in the French system, I also felt that there was a greater cultural weight and institutionalized preference (via course requirements and choice of content) for theoretical topics in ML compared to US universities.
I know little about ENS, but somewhat doubt that it’s as significantly different of an experience from US/UK counterparts.
Alternative model: French mathematicians don’t overperform in an objective sense. Rather, French mathematicians happened to end up disproportionately setting fashion trends in pure mathematics for a while, for reasons which are mostly just signalling games and academic politics rather than mathematical merit.
The Bourbaki spring to mind here as a central example.
Fwiw, the French dominance isn’t confined to Bourbakist topics. E.g. Pierre Louis Lions won one of the French medals and is the world most cited mathematician, with a speciality in PDEs. Some of his work investigates the notion of general nonsmooth (“viscosity”) solutions for the general Hamilton-Jacobi(-Bellmann) equation both numerically and analytically. It’s based on a vast generalization of the subgradient calculus (“nonsmooth” calculus), and is very directly related to good numerical approximation schemes.
Maryna Viazovska, the Ukrainian Fields medalist, did her PhD in Germany under Don Zagier. ---
I once saw at least one French TV talkshow where famous mathematicians were invited (I don’t find the links anymore). Something like that would be pretty much unthinkable in Germany. So I wonder if math has generally more prestige in France than e.g. in Germany.
One point evoked by other comments, which I’ve realized only after leaving France and living in the UK, is that there is still a massive prestige for engineering. ENS is not technically an engineering school, but it benefits from this prestige by being lumped with them, and by being accessed mainly from the national contests at the end of Prepas.
As always with these kind of cultural phenomena, I didn’t really notice them until I left France for the UK. There is a sense in France (more when I was a student, but still there) that the most prestigious jobs are engineering ones. Going to engineering school is considered one of the top options (with medecine), and it is considered a given that any good student with a knack for maths, physics, science, will go to prepa and engineering school.[1] It’s almost free (and in practice is free if your parents don’t make more than a certain amount), and it is guaranteed to lead to a good future.
This means that the vast majority of mathematical talent studies the equivalent of a undergraduate degree in maths, compressed in the span of 2 years. In addition of giving the standard french engineer much more of a mathematical training, it shows to the potential mathematicians, by default, a lot of what they could do. And if they decide to go to ENS (or Polytechnique, which is the best engineering school but still quite researchy if you want to), this is actually one of the most prestigious options you could take.
Similarly, the prestige of engineering (and science to some extent) impacts what people decide to do after their degrees. I remember that in my good prepa and my good engineering school, the cool ones were those going to build planes and bridges. The ones who went into consulting and finance were pitied and mocked as the failures, not the impressive successes to emulate. Yet what my UK friends tell me is that this is the exact opposite of what happens even in great universities in the UK.
This has become less true, as more private schools open, and the whole elitist system is wormed out by software engineering startups (which generally doesn’t ask you for an engineering degree, as opposed to the older big french companies).
I’m not 100% sure about the second factor but the first is definitely a big factor. There’s no institution which is more dense in STEM talent than ENS to my knowledge, and elites there are extremely generalist compared to equivalent elites I’ve met in other countries like the US (e.g. MIT) for instance. The core of “Classes Préparatoires” is that it pushes even the world best people to grind like hell for 2 years, including weekends, every evenings etc.
ENS is the result of: push all your elite to grind like crazy for 2 years on a range of STEM topics, and then select the top 20 to 50.
Maybe I’m going crazy, but the frequent use of qualifiers for almost every noun in your writing screams of “LLM” to me. Did you use LLM assistance? I don’t get that same feel from your comments, so I’m learning toward an AI having written only the Shortform itself.
If you did use AI, I’d be in favor of you disclosing that so that people like me don’t feel like they’re gradually going insane.
If not, then I’m sorry and retract this. (Though not sure what to tell you—I think this writing style feels too formal and filled with fluff like “crucial” or “invaluable”, and I bet you’ll increasingly be taken for an AI in other contexts.)
Yes I use LLMs in my writing [not this comment] and I strongly encourage others to do so too.
This the age of Cyborgism. Jumping on making use of the new capabilities opening up will likely be key to getting alignment right. AI is coming, whether you like it or not.
There is also a mundane reason: I have an order of magnitude more ideas than I can write down. Using LLMs allows me to write an essay in 30 min which otherwise would take half a day.
Oh yeah no problem with writing with LLMs, only doing it without disclosing it. Though I guess this wasn’t the case here, sry for flagging this.
I’m not sure I want to change my approach next time though, bc I do feel like I should be on my toes. Beware of drifting too much toward the LLM’s stylebook I guess.
If the same pattern of other innovations hold, then it seems more likely that it is more a questions of concentration in one place (for software: Silicon Valley). Maybe one French university, maybe the École Normale Supérieure, managed to become the leading place for math and now every mathematician tries to go there.
I wonder about the extent to which having an additional level of selection helps.
High school curricula are generally limited by having to be able to be taught by a large number of teachers all around the country and by needing a minimum number of students at the school who are capable of the content.
If the préparatoires can put more qualified teachers and students together that would allow significant development and running selection for elite universities after such an intermediate preparatory program it would reduce the chance that talented students aren’t missed due to having attended a high school that is weaker at maths (even though it sounds like the preparatories have a selection bar too, I assume it’s quite a bit lower than performing well enough to get into a top institution).
I agree with the previous points, but I would also add historical events that led to this. Pre-WW I Germany was much more important and plays the role that France is playing today (maybe even more central), see University of Göttingen at the time.
After two world wars the German mathematics community was in shambles, with many mathematicians fleeing during that period (Grothendieck, Artin, Gödel,...). The university of Bonn (and the MPI) were the post-war project of Hirzebruch to rebuild the math community in Germany.
I assume France then was then able to rise as the hotspot and I would be curious to imagine what would have happened in an alternative timeline.
Why Do the French Dominate Mathematics?
France has an outsized influence in the world of mathematics despite having significantly fewer resources than countries like the United States. With approximately 1/6th of the US population and 1/10th of its GDP, and French being less widely spoken than English, France’s mathematical achievements are remarkable.
This dominance might surprise those outside the field. Looking at prestigious recognitions, France has won 13 Fields Medals compared to the United States’ 15 a nearly equal achievement despite the vast difference in population and resources. Other European nations lag significantly behind, with the UK having 8, Russia/Soviet Union 6⁄9, and Germany 2.
France’s mathematicians are similarly overrepresented in other mathematics prizes and honors, confirming this is not merely a statistical anomaly.
I believe two key factors explain France’s exceptional performance in mathematics while remaining relatively average in other scientific disciplines:
1. The “Classes Préparatoires” and “Grandes Écoles” System
The French educational system differs significantly from others through its unique “classes préparatoires” (preparatory classes) and “grandes écoles” (elite higher education institutions).
After completing high school, talented students enter these intensive two-year preparatory programs before applying to the grandes écoles. Selection is rigorously meritocratic, based on performance in centralized competitive examinations (concours). This system effectively postpones specialization until age 20 rather than 18, allowing for deeper mathematical development during a critical cognitive period.
The École Normale Supérieure (ENS) stands out as the most prestigious institution for mathematics in France. An overwhelming majority of France’s top mathematicians—including most Fields Medalists—are alumni of the ENS. The school provides an ideal environment for mathematical talent to flourish with small class sizes, close mentorship from leading mathematicians, and a culture that prizes abstract thinking.
This contrasts with other countries’ approaches:
Germany traditionally lacked elite-level mathematical training institutions (though the University of Bonn has recently emerged as a center of excellence)
The United States focuses on mathematics competitions for students under 18, but these competitions often emphasize problem-solving skills that differ significantly from those required in mathematical research
The intellectual maturation between ages 18 and 20 is profound, and the French system capitalizes on this critical developmental window.
2. Career Stability Through France’s Academic System
France offers significantly more stable academic positions than many other countries. Teaching positions throughout the French system, while modestly compensated, effectively provide tenure and job security.
This stability creates an environment where mathematicians can focus on deep, long-term research without the publish-or-perish pressure common in other academic systems. In mathematics particularly, where breakthroughs often require years of concentrated thought on difficult problems, this freedom to think without immediate productivity demands is invaluable.
While this approach might be less effective in experimental sciences requiring substantial resources and team management, for mathematics—where the primary resource is time for thought—it has proven remarkably successful.
I don’t buy your factors (1) or (2). Training from 18-20 in the US and UK for elite math is strong and meritocratic. And brilliant mathematicians have career stability in the US and UK.
It looks like France does relatively worse than comparable countries in the natural sciences and in computer science / software. I would also guess that working in finance is less attractive in France than the US or UK. So one possible factor is opportunity cost.
https://royalsocietypublishing.org/doi/10.1098/rsos.180167
Those are some good points certainly.
The UK/US system typically gives tenure around ~40, typically after ~two postdocs and a assistant → associate → full prof.
In the French system a typical case might land an effectively tenured job at 30. Since 30-40 is a decade of peak creativity for scientists in general, mathematicians in particular I would say this is highly
Laurent Lafforgue is a good example. Iirc he published almost nothing for seven years after his PhD until the work that he did for the Fields medal. He wouldnt have gotten a job in the American system.
He is an extreme example but generically having many more effectively tenured positions at a younger age means that mathematicians feel the freedom to doggedly pursue important, but perhaps obscure-at-present, research bets.
My point is primarily that the selection is at 20, instead of at 18. It s not about training per se, although here too the French system has an advantage. Paris has ~ 14 universities, a number of grand ecolees, research labs, etc a large fraction which do serious research mathematics. Paris consequently has the largest and most diverse assortiment of advanced coursework in the world. I don’t believe there is any place in the US that compares [I’ve researched this in detail in the past].
The UK does not have the same tenure system as the US. I believe top mathematicians have historically (i.e. last 70 years) often become permanent lecturers fairly young (e.g. by age 32).
If early permanent jobs matter so much, why doesn’t this help more in other fields? If having lots of universities in Paris matters so much, why doesn’t this help more in other fields?
I wouldn’t claim to be an expert on the UK system but from talking with colleagues at UCL it seems to be the case that French positions are more secure and given out earlier [and this was possibly a bigger difference in the past]. I am not entirely sure about the number 32. Anecdotally, I would say many of the best people I know did not obtain tenure this early. This is something that may also vary by field—some fields are more popular, better funded because of [perceived] practical applications.
Mathematiscs is very different from other fields. For instance: it is more long-tailed, benefits from ′ deep research, deep ideas’ far more than other fields, is difficult to paralellize, has ultimate ground truth [proofs], and in large fraction of subfields [e.g. algebraic geometry, homotopy theory …] the amount of prerequisite knowledge is very large,[1] has many specialized subdisciplines , there are no empirical
All these factors suggest that the main relevant factor of production is how many positions that allow intellectuall freedom, are secure, at a young age plus how they are occupied by talented people is.
e.g. it often surprises outsiders that in certian subdisciplines of mathematics even very good PhD students will often struggle reading papers at the research frontier—even after four years of specialized study.
I’d defend a version of claim (1): My understanding is that to a greater extent than anywhere else, top French students wanting to concentrate in STEM subjects must take rigorous math coursework from 18-20. In my one year experience in the French system, I also felt that there was a greater cultural weight and institutionalized preference (via course requirements and choice of content) for theoretical topics in ML compared to US universities.
I know little about ENS, but somewhat doubt that it’s as significantly different of an experience from US/UK counterparts.
Certainly for many/most other subjects the French system is not so good. E.g. for ML all that theory is mostly a waste.
This is the first time I’ve heard this claim. Any background/cites I should look into for this?
Alternative model: French mathematicians don’t overperform in an objective sense. Rather, French mathematicians happened to end up disproportionately setting fashion trends in pure mathematics for a while, for reasons which are mostly just signalling games and academic politics rather than mathematical merit.
The Bourbaki spring to mind here as a central example.
Sure happy to disagree on this one.
Fwiw, the French dominance isn’t confined to Bourbakist topics. E.g. Pierre Louis Lions won one of the French medals and is the world most cited mathematician, with a speciality in PDEs. Some of his work investigates the notion of general nonsmooth (“viscosity”) solutions for the general Hamilton-Jacobi(-Bellmann) equation both numerically and analytically. It’s based on a vast generalization of the subgradient calculus (“nonsmooth” calculus), and is very directly related to good numerical approximation schemes.
Maryna Viazovska, the Ukrainian Fields medalist, did her PhD in Germany under Don Zagier.
---
I once saw at least one French TV talkshow where famous mathematicians were invited (I don’t find the links anymore). Something like that would be pretty much unthinkable in Germany. So I wonder if math has generally more prestige in France than e.g. in Germany.
One point evoked by other comments, which I’ve realized only after leaving France and living in the UK, is that there is still a massive prestige for engineering. ENS is not technically an engineering school, but it benefits from this prestige by being lumped with them, and by being accessed mainly from the national contests at the end of Prepas.
As always with these kind of cultural phenomena, I didn’t really notice them until I left France for the UK. There is a sense in France (more when I was a student, but still there) that the most prestigious jobs are engineering ones. Going to engineering school is considered one of the top options (with medecine), and it is considered a given that any good student with a knack for maths, physics, science, will go to prepa and engineering school.[1] It’s almost free (and in practice is free if your parents don’t make more than a certain amount), and it is guaranteed to lead to a good future.
This means that the vast majority of mathematical talent studies the equivalent of a undergraduate degree in maths, compressed in the span of 2 years. In addition of giving the standard french engineer much more of a mathematical training, it shows to the potential mathematicians, by default, a lot of what they could do. And if they decide to go to ENS (or Polytechnique, which is the best engineering school but still quite researchy if you want to), this is actually one of the most prestigious options you could take.
Similarly, the prestige of engineering (and science to some extent) impacts what people decide to do after their degrees. I remember that in my good prepa and my good engineering school, the cool ones were those going to build planes and bridges. The ones who went into consulting and finance were pitied and mocked as the failures, not the impressive successes to emulate. Yet what my UK friends tell me is that this is the exact opposite of what happens even in great universities in the UK.
This has become less true, as more private schools open, and the whole elitist system is wormed out by software engineering startups (which generally doesn’t ask you for an engineering degree, as opposed to the older big french companies).
I’m not 100% sure about the second factor but the first is definitely a big factor. There’s no institution which is more dense in STEM talent than ENS to my knowledge, and elites there are extremely generalist compared to equivalent elites I’ve met in other countries like the US (e.g. MIT) for instance. The core of “Classes Préparatoires” is that it pushes even the world best people to grind like hell for 2 years, including weekends, every evenings etc.
ENS is the result of: push all your elite to grind like crazy for 2 years on a range of STEM topics, and then select the top 20 to 50.
Maybe I’m going crazy, but the frequent use of qualifiers for almost every noun in your writing screams of “LLM” to me. Did you use LLM assistance? I don’t get that same feel from your comments, so I’m learning toward an AI having written only the Shortform itself.
If you did use AI, I’d be in favor of you disclosing that so that people like me don’t feel like they’re gradually going insane.
If not, then I’m sorry and retract this. (Though not sure what to tell you—I think this writing style feels too formal and filled with fluff like “crucial” or “invaluable”, and I bet you’ll increasingly be taken for an AI in other contexts.)
Yes I use LLMs in my writing [not this comment] and I strongly encourage others to do so too.
This the age of Cyborgism. Jumping on making use of the new capabilities opening up will likely be key to getting alignment right. AI is coming, whether you like it or not.
There is also a mundane reason: I have an order of magnitude more ideas than I can write down. Using LLMs allows me to write an essay in 30 min which otherwise would take half a day.
Oh yeah no problem with writing with LLMs, only doing it without disclosing it. Though I guess this wasn’t the case here, sry for flagging this.
I’m not sure I want to change my approach next time though, bc I do feel like I should be on my toes. Beware of drifting too much toward the LLM’s stylebook I guess.
“Utter elitism” is a nice article about this phenomenon
If the same pattern of other innovations hold, then it seems more likely that it is more a questions of concentration in one place (for software: Silicon Valley). Maybe one French university, maybe the École Normale Supérieure, managed to become the leading place for math and now every mathematician tries to go there.
I wonder about the extent to which having an additional level of selection helps.
High school curricula are generally limited by having to be able to be taught by a large number of teachers all around the country and by needing a minimum number of students at the school who are capable of the content.
If the préparatoires can put more qualified teachers and students together that would allow significant development and running selection for elite universities after such an intermediate preparatory program it would reduce the chance that talented students aren’t missed due to having attended a high school that is weaker at maths (even though it sounds like the preparatories have a selection bar too, I assume it’s quite a bit lower than performing well enough to get into a top institution).
I agree with the previous points, but I would also add historical events that led to this.
Pre-WW I Germany was much more important and plays the role that France is playing today (maybe even more central), see University of Göttingen at the time.
After two world wars the German mathematics community was in shambles, with many mathematicians fleeing during that period (Grothendieck, Artin, Gödel,...). The university of Bonn (and the MPI) were the post-war project of Hirzebruch to rebuild the math community in Germany.
I assume France then was then able to rise as the hotspot and I would be curious to imagine what would have happened in an alternative timeline.