> but it now requires so much knowledge just to get to the frontiers of human knowledge, not to speak of making a dent into uncharted territory, that results are being obtained later and later in life. When mathematicians have had time to accrue sufficient knowledge while still being sharp enough to make the intellectual leap.
The sad part is that as the trend continues we may reach a point where a mathematician's intellectually productive life is not sufficient to contribute anything novel, statistically speaking.
IMO, an under appreciated dynamic across the board of human endeavours.
With increasing complexity comes the need for more time to understand and master anything.
That's a very big claim, perhaps true only if human endeavor only ever builds linearly. But I dont think that is true. Certainly not in the arts or music, e.g. the Beatles did not need to ingest the entire corpus of Mozart or even Scott Joplin to be highly productive. Although, it certainly helped that they were extremely open minded to all types of music.
A second example is that of recent advances in virtualization, where the last decade of advances in things like cgroups, namespaces and containers were all done by people who I assert had no training in IBM MVS/zOS and therefore weren't building on what went before (imho, to their detriment).
The Beatles themselves didn't need to ingest the entire corpus of past musicians to be productive & inventive, but they did stand on the shoulders of giants musically (vs being born in some prior century), and they were in the right time & place to be part of a growing scene whose smarts exceeded that of any of it's individual participants. This is the nature of 'scenius' and of golden ages - a lot of ideas are already teed up in the collective consciousness.
> only if human endeavor only ever builds linearly.
A lot of mathematics is cumulative, though.
And even if you can go further by going thinner (specialising more), maybe breakthroughs require lateral thinking and connections that are predicated on not being too specialised, but having breadth also. If that is the case, sooner or later we might be in trouble.
It's also a major failure in didactics. It feels like very little of the new knowledge since twentieth century has been truly digested for easy teaching. Why isn't general relativity taught in elementary school? It should be possible.
FWIW I notice how remarkably bad I was taught math at school.
When I started to go to university I really noticed how bad it was. At university the jump forward was really noticeable.
For example, at school they would show you a couple of simple explanations about derivative math or integrals, briefly and start with all the formulas.
At university I used to have a teacher that started with: history of mathematics, why they were invented, made a point about its primarily practical origins.
To explain things, he could most of the time come with real-life instances of application and there were much more often intuitive or geometric interpretations of the techniques used much more often even before starting the explanation itself to have an intuitive idea and visualization of what you were achieving.
After that, I noticed that to learn math, the first thing is to develop an intuitive, non-mathy idea of what you are doing and later formalize it.
At school and high school they just taught it as almost-memorize tables, apply formulas.
This! Identical situation in the UK from my experience. Why maths is taught completely separated from its history and its purpose is baffling. Well actually it’s not: it’s because it serves the teachers. You hammer the formulas and patterns into the kids so they can pass the exams. Your school gets good grades and the principal is happy because he can now market his school as successful and the government inspectors are happy because the school is hitting its metrics. Meanwhile, the kids haven’t actually learned anything. As soon as the exam is passed they forget the formulas and go on with their lives.
We should be explaining the story of maths and how it benefitted society. We should be asking kids about their interests and then showing them how mathematical tools can be used in those areas. We need to show kids how maths is tied to real life rather than just presenting them with a boring formulas to memorise.
I don't think this is the case. I think it's because the teachers themselves don't have a good grasp on "the story of maths and how it benefitted society".
There is a silly meme about asking high school maths teachers "how will we use this in life", and imo it's not because there isn't a good response, but rather that it requires a good understanding of the ways math is actually used. Few high school teachers have actually themselves used the math they teach for anything other than academic exercise. Someone trained in control theory or using physics equations can make things that appear almost magical using maths, and if they are talented, they can find a way to explain it to laypeople. However, people with that combination of talents are desired by just about everybody, from universities to companies, and high schools simply have no way to compete (not least because teaching high schoolers is a soul-crushing job for bureaucratic reasons)
Yeah this is a very good point. The amount of people who basically go from school to university to school teacher is alarmingly high and bad for society in my opinion. Grown adults who have never known anything but the school system teaching the next generation. It’s ridiculous. I do still think the original point about hitting metrics is a major contributing factor though along with low pay and underfunding.
When I came across the (obvious, in retrospect) visual explanation for why (a + b)(a + b) == a^2 + 2ab + b^2 I was blown away by how simple it was, and by why on earth I had to just memorise that formula in school.
It's interesting that some people find that way easier to remember. For me, multiplying out the terms seems faster, probably only because I've practised so many times in my life that I can picture the algebra in my head. It also seems more general, as it's a technique which you have to know anyway. I guess it comes down to being more geometrically minded vs algebraically minded. School should try to cater to both!
I see no reason to believe that general relativity could be taught in elementary school. It requires advanced undergraduate or graduate mathematics, not to mention that the physics itself is quite difficult (to put it mildly).
To put the blame on didactically seems to miss the more important factor that humans just aren’t that intelligent, save the one in a million genius who might have the intellectual capacity to learn something so difficult at such a young age.
Interesting! As other comments ts have pointed out, didactics as an important aspect of how humanity responds to such a dynamic.
Otherwise, I’d say our two thoughts are connected. With increasing difficulty in understanding new progress, there could be an inertial tendency to over emphasise the importance of old knowledge because it’s comforting/easier/pragmatic for teachers and parents.
Agreed, it feels like the days of maverick engineers leading huge efforts is mostly over. Will there be any more Kelley Johnsons, or Gene Kranzs in our lifetimes?
Engineering is different from math. A large part of the reason why engineering innovation is lacking in a particular field is due to entrenched players, heavy bureaucracy, and government regulation, for instance with spaceflight and EVs.
Tools can be used for this purpose. New tools can unlock new abilities for engineers. I’m not sure if it applies to math but I don’t have any reason why it shouldn’t.
This sentiment is humorous, as I'm an optimist apparently. More children will learn more quickly. Applying AI to education should supercharge the smartest.
I don't know but it also fails to disregard that the smartest people (whatever that means) can have executive function issues that are orthogonal to their mental capacity.
Additionally, the belief that intelligence alone will improve the world is misguided imo. You need empathetic, intelligent people.
You must be an optimist if you think what passes for AI right now could in any way help education. It's an incredibly useful tool, but not in the way you almost certainly are thinking.
IMO, an under appreciated dynamic across the board of human endeavours.
With increasing complexity comes the need for more time to understand and master anything.