(...) the term technical is a red flag for me, as it is many times used not for the routine business of implementing ideas but for the parts, ideas and all, which are just hard to understand and many times contain the main novelties.
- Saharon Shelah
As a true-born Dutchman I endorse Crocker's rules.
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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.
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.
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.
Certainly for many/most other subjects the French system is not so good. E.g. for ML all that theory is mostly a waste.
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].
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.
1ADHD is about the Voluntary vs Involuntary actions
The way I conceptualize ADHD is as a constraint on the quantity and magnitude of voluntary actions I can undertake. When others discuss actions and planning, their perspective often feels foreign to me—they frame it as a straightforward conscious choice to pursue or abandon plans. For me, however, initiating action (especially longer-term, less immediately rewarding tasks) is better understood as "submitting a proposal to a capricious djinn who may or may not fulfill the request." The more delayed the gratification and the longer the timeline, the less likely the action will materialize.
After three decades inhabiting my own mind, I've found that effective decision-making has less to do with consciously choosing the optimal course and more with leveraging my inherent strengths (those behaviors I naturally gravitate toward, largely outside my conscious control) while avoiding commitments that highlight my limitations (those things I genuinely intend to do and "commit" to, but realistically never accomplish).
ADHD exists on a spectrum rather than as a binary condition. I believe it serves an adaptive purpose—by restricting the number of actions under conscious voluntary control, those with ADHD may naturally resist social demands on their time and energy, and generally favor exploration over exploitation.
Society exerts considerable pressure against exploratory behavior. Most conventional advice and social expectations effectively truncate the potential for high-variance exploration strategies. While one approach to valuable exploration involves deliberately challenging conventions, another method simply involves burning bridges to more traditional paths of success.
3
1I use LLMs throughout my personal and professional life. The productivity gains are immense. Yes hallucination is a problem but it's just as spam/ads/misinformation on wikipedia/internet - an small drawback that doesn't oblivate the ginormous potential of the internet/LLMs
I am 95% certain you are leaving value on the table.
I do agree straight LLMs are not generally intelligent (in the sense of universal intelligence/AIXI) and therefore not completely comparable to humans.
This was basically my model since i first started paying attention to modern AI
Curious why did you think differently before ? :)
Cephalopods are highly intelligent but also have a very short lifecycle and very large number of offspring - making them a prime target for a artificial breeding program for intelligence uplifting.
E.g. the lifecycle of many octopi is about 1-2 years. A 15 year program could potentially breed for very significant behaviourial changes.
For instance, dog and horse breeds can be made within 10 generations.
Google says the female giant pacific octopis lays between 120k and 400k eggs at the end of her life. Giant pacific octopi live about 3-5 years.
I asked Claude to make some back of the enveloppe calculations
>>>I'll use the breeder's equation to estimate the potential response to selection for intelligence in octopuses. The breeder's equation states:
R = h² × S
Where:
Let's make some reasonable assumptions for these parameters in octopuses:
Let's calculate the response to selection per generation:
R = 0.4 × 3 = 1.2 standard deviations per generation
For a breeding program over 15 years with Giant Pacific Octopuses (3-5 year lifecycle):
For octopuses with a shorter 1-2 year lifecycle:
This represents a substantial shift in the distribution of intelligence. For context, the difference in IQ between average humans and those considered profoundly gifted is about 4 standard deviations. A shift of 12 standard deviations would be extraordinary.
However, several factors would likely limit these theoretical gains:
Even with these limitations, the potential for significant intelligence enhancement in octopuses through selective breeding appears substantial, especially for species with shorter lifecycles.
Alexander reacting to Claude back of enveloppe calculation:
Narrow sense heritability is probably higher for intelligence. Indeed, at this level of selection one would have to worry about inbreeding depression, selection plateaus etc
My best guess is that 6 SD would be possible IF one was actually able to accurately select top 0.1%.
This is probably quite hard. GWAS data for humans currently does not allow for this kind of precision. Doing the selection accurately / i.e. estimating the gradient is the main rate-limiting step [as it is in deep learning!]. One would need to construct psychometrically valid tests for cephalopod, run them at scale.
How much is 6 SD? Octopi might be about as intelligent as a dog (uncertain about this). 6 SD would be quite insane, and would naively plausibly push them tonthe upper end of dolphin/chimpanzee intelligence. The main limitor in my mind is that octopi are not natively social species and are do not have a long enough lifecycle to do significant learning so this might not actually lead to intelligence uplifting. Additionlly, there is the issue that octopi dont have vocal chords so would need to communicate differently.
Ofc like the famous Soviet silver fox breeding program one could separately select for sociability [which might be more important for effective intelligence. Iirc wolfs usually outperform dogs cognitively yet some dog breeds are generally considered more intelligent in a relevant sense.]