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Comment by pa7x1 | Hacker News Reader

Math used to be a young person game, 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. And as population seems to be close to peaking, we will also have less chances of exploring the extremes of mathematical dexterity.

Perhaps we could then rely on computer assisted theorem provers. Or life extension, as long as intellectually productive years are also increased. Or we will need to focus and specialize kids earlier on.

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> [...] 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.

That's not really the case here. Zhang spent 10 years out of school working in fields in the cultural revolution, and didn't start college until he was 23. After his PhD, he couldn't get an academic job for 8 years and ended up delivering food and working as an accountant. He only got a part time lectureship after that. He wasn't made a professor until his big proof at the age of 58.

It seems incredibly likely he'd have been able to proove his big proofs faster had his life gone otherwise.

> It seems incredibly likely he'd have been able to proove his big proofs faster had his life gone otherwise.

Why do you think that? It takes sometimes years (!) until you get used tona certain mathematical theory. And how long it takes isa very personal thing.

Remember von Neumann, who that that we don't learn nu mathematics, but only get used to it.

I think the only thing one can say was incredibly likeli is that Zhang, while not formally employed as a mathematician, kept thinking about mathematics (a rather common for people with mathematical training, though many just stick to do occasional problem solving).

> Zhang spent 10 years out of school working in fields in the cultural revolution, and didn't start college until he was 23. After his PhD, he couldn't get an academic job for 8 years and ended up delivering food and working as an accountant.

During that time he definitely spend less time on maths than he could or would in a better situation.

Especially cultural revolution oppression was monumental waste of human lives. See https://en.wikipedia.org/wiki/Cultural_Revolution#Death_toll alone

A self-made man :)

> 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.

This sentiment is humorous, as I'm an optimist apparently. More children will learn more quickly. Applying AI to education should supercharge the smartest.

> Applying AI to education should supercharge the smartest.

What does that mean?

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.

As we do not yet have artificial intelligence making confident statements about what its effects might be is surely rather rash.

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.

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?

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.

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.

There is still plenty of amazing engineering going on, for example EUV development at ASML.

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.

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.

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.

Talking about Spain, btw.

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.

This is exactly the kind of intuitions I am talking about. Much easier to explain like that.

By "visual explanation" do you mean imagining a square whose sides are length (a + b), and breaking it up into smaller squares and rectangles?

Or do you mean mulitplying out the terms:

  (a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2

?

The former, yes.

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 wouldn't use it to calculate, but I would use it to explain why the calculation is what it is.

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.

> because it serves the teachers

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.

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.

"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."

People talk about this a lot. While I think it could happen for certain subdisciplines (it already takes essentially an entirely PhD's worth of time to learn all the necessary background to be an algebraic geometer, so most algebraic geometry PhD students publish nothing besides their thesis during their PhD studies), it can never happen to mathematics as a whole. If one part of math gets too deep, you can always go somewhere else, where the water is still "shallow."

I suspect computers can also help us get deeper. Stuff like computer algebra systems.

Maybe some CAS-assisted work gets us into feedback loops allowing us to go indefinitely, as in a technological singularity.

But the "shallow" part is also quite wide.

You can teach people what you've learned forever, for instance.

do you have experience with CAS? i would love to learn how to use CAS to write proof more effectively

CAS can help with the more mechanical parts of a proof. I use it often to quickly check if something can be rewritten to something else I want. The sad part is that even if the CAS doesn't find a solution it doesn't mean there is no solution. It just didn't find it. But it can save a lot of work if the first thing you do is just quickly check, if you're lucky you just saved yourself a lot of work.

I don't know what field you are in so it may or may not be helpful to you.

One very common and simple use case is looking for counter examples. If your theory states for example that "all Matrices with property X also have property Y" then it is quick and easy to generate a whole bunch of matrices with property X and check that they all have property Y. Of course that doesn't actually prove anything, but it can be used to disprove a statement and save you a bunch of time chasing down a dead end.

To be honest, my experience is limited to double-checking my algebra with the free Wolfram Alpha. I need it maybe a few times a year.

I’m not so sure. The same argument would apply to theoretical physics in 1960. Circa 2023, there are remarkably few shallow parts of physics.

Math as a whole may last longer, but this list reminds us how far we’ve come in a mere few millennia: https://usercontent.irccloud-cdn.com/file/SaI50Q1d/166786520...

On the timescale of civilization, it seems less and less likely that lone mathematicians can revolutionize the field.

We’re fortunate to have been born so early, relatively speaking.

Physics is limited by having to represent phenomena in our physical world simply.

Mathematics is not just a small integer multiple larger than this.

This is almost correct.

It's not that math is vastly larger (or more sophisticated) as a field. Both fields are infinitely large in many senses. Rather, the number of respectable starting points where you can do interesting things is much larger, orders larger in math.

There are plenty of unexplored things in physics too; the key word is "respectable".

Looking from the outside, physics suffers a lot from fashion/hot trend tendencies, where you need to be doing the "hot" thing to make the jumps necessary to the coveted Tenure Track — and otherwise, you get kicked out.

> it seems less and less likely that lone mathematicians can revolutionize the field.

Which inspires the question: how much can cutting edge math be parallelized?

But you can manufacture new areas of mathematics. For example, Conway's Game of Life, and then prove theorems on it.

> If one part of math gets too deep, you can always go somewhere else, where the water is still "shallow."

Yes, but the shallow areas aren't very interesting, which is why people work in the deep areas.

Most of the now-deep, now-interesting areas were once shallow and uninteresting.

Then you spend your whole PhD reinventing a wheel that has a different name in the 30 year old textbook from the next field over, and neither your peers or professor have any awareness of that

Or, as Juergen Schmidhuber likes to point out, in the same field.

There's also occasionally realignments where the deep stuff gets shallower. It takes a lot of rickety scaffolding to get to a new place, and occasionally the finished product stands fine on its own.

The simplest example that comes to mind is that you can learn group theory without really needing to know anything about Galois theory. I also imagine there's a lot of good math that has shed vestigial physics...

Isn't this evidence that this process has started to occur? The difficulty to make progress in certain areas of math pushes mathematicians to the shallower areas where it's easier to make a contribution. Progress in the first ones will stall and at some point the shallower areas will become less shallow and same phenomenon will occur.

Maybe we will keep forever discovering new shallow areas but I suspect this is not the case. In any case this is a phenomenon that I think will play out in the next few hundred years, not sufficiently impactful in the next few decades but more and more noticeable.

Another approach to the finite-mathematician-lifespan problem might be to develop new foundations that are closer to the edge. I expect there's a logical universe in which sets are bizarre and hard to construct but objects which would take a modern mathematician years to grok are convenient and simple to work with.

Homotopy type theory?

My favorite thought game along these lines is imagining an alien civilization where permutations are the fundamental object instead of the natural numbers.

Maybe food is plentiful enough that they never need to bother with its quantity, but there's some strobing energy source in their environment which periodically walks important chemicals through an ultimately cyclical series of states.

So like, the food is poisonous on strobes 3 and 8 of a 14 strobe cycle, and understanding that is the key to staying alive in their environment.

This approach reminds what the geometric algebra folks are trying to do.

>Math used to be a young person game, 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

>The sad part is that as the trend continues we may reach a point where a mathematician's intellectually productive life is not sufficient

Hmm. I had never thought of it like this. Is it possible for human knowledge to become so advanced in a single subject that it takes a persons entire life to learn just one subject? That's already true of something like the human body, which is why doctors specialize in organs or regions of the body. The more human knowledge expands, the more individuals specialize into increasingly smaller niches.

This is a growing problem in many fields, IMHO. I've been wondering for awhile if it's an inherent flaw in knowledge in that if knowledge can't supplant older knowledge in a high compressed reduced form as things progress, we're just building so much information/knowledge for any given field that at some point, it may be quicker to simply rediscover the process than to search the knowledge for the prior work.

This is where longevity research a la Harold Katcher comes in and allows for super centenarian geniuses to shape the future of humanity.

But what if there are necessary prerequisites for making significant advances in longevity, which themselves are so complex that they require more than a human lifespan to master? We could be at a dead end unless the computers start doing the work. And even then, perhaps the same impediment applies to the development of an AI that can bootstrap other AIs into getting the job done.

Harold Katcher: “The Illusion of Knowledge”.

If you could live two or three lifetimes, would you really want to spend them all on the same thing?

Feels like attention would become the limiting factor.

Even with optimal compression, there is still a finite minimum size.

Here is a short story that covers that idea: https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/

Thanks for posting it!

I've read it, lost it, and have been searching for it for a few years now.

I think we're kind of becoming multi a cell organism.

There's limit to human knowledge, but we are getting more and more efficient at communication, both with other humans and with machines.

Even without something like Neuralink we are creating abstractions and interfaces allowing us to quickly connect our work with others. E.g. especially with ML you can lazy load all necessary context.

Think of it like Rust eventually supplanting C++ once the ecosystem and libraries are complete enough for people to make the change.

The cognitive burden reduces via the new foundations not requiring deep understanding of the old ones to be able to make new steps forward.

This is not a good example, as Rust does not build on C++, it simply replaces it as the foundation. Of course, the development of Rust is built upon lessons learned from C++.

But there is no universal law that says that the foundations of a field must be simple enough for one human to understand them in 70 years. It may well be that the simplest possible statement of a field of knowledge is still too complex for a single human to understand it in a normal life-span (not to mention that our capacity for storing information is limited - you can't actually continually learn new things for 70 years without forgetting much of what you learned initially).

Even today, you could spend literally your entire life trying to learn everything we know about the human body and you would almost certainly die before having learned everything. Now, fortunately, there is plenty of real work, both as a doctor and as a researcher, that can be done by focusing on just one aspect of the body (say, the circulatory system) and having only relatively shallow knowledge about the rest. Still, there is the possibility already that a mind that could build on all of the deep knowledge we have could come up with new ideas in medicine and biotech that we are unable to because of this silo-ing.

> This is not a good example, as Rust does not build on C++, it simply replaces it as the foundation.

No, that's exactly the point. You can thereby drop the cognitive load of C++ and build greenfield on the Rust foundations.

While I get what you're saying, and clearly there is "more stuff", I think we shouldn't discount how progress can make things simpler on this front. Proofs we learn are usually much more refined than the original versions. Notational improvements and more interesting abstractions mean we can cover more ground more quickly.

Another angle here, I remember seeing a study where some children were just... taught algebra. Like given high school algebra classes in 3rd grade, and kids were able to absorb all that abstract reasoning "just fine" (according to the study).

Of course there's only so much abstraction that can be done, but I think we shouldn't assume we are at the end of history on much of anything (except for parsing algorithms)

To add to your comment, quaternions predate rotational matrix operators by a considerable amount (1843 vs not exactly clear, ~1900 with Peano or ~1920 with Weyl), despite quaternions being much more challenging to manipulate. There are definitely simpler ways to view the same things.

There was a cottage industry of exotic hypercomplex numbers that disappeared when linear algebra matured to eclipse them.

In fact, Maxwell's Equations were originally derived with quaternions.

I am somewhat unconvinced that quaternions are more challenging, or at least a worse way to think about the issue :

https://eater.net/quaternions

And speaking of Maxwell's Equation"s" :

http://www.av8n.com/physics/maxwell-ga.htm#sec-preview

Well, a rotation matrix doesn't require doing 2 half rotations, and doesn't require reaching into the 4th dimension in such a way that it gets perfectly cancelled out. It doesn't require abstract analogies about cubes with strings glued to them or people holding coffee cups.

With some familiarity with linear algebra, it's easy to derive the formula for constructing a rotation matrix. You just have to think about what the operation does to the axes. The derivation for quaternion rotation is far more abstract, by virtue of the operation we actually care about involving a sandwich of multiplications with unclear 4 dimensional meaning. There's no hyperspheres with a rotation matrix.

Augmenting your space to handle not just rotations & scaling, but translations is easy for matrices, just requires a homogeneous coordinate and you get 4x4 matrices with intuitive columns.

Augmenting quaternions to handle translations requires the 8 dimensional dual-quaternions.

I definitely like geometric algebra, it's a very nice continuation of topics in linear algebra and makes it clear why things like normals behave differently from standard vectors. But I don't use it every day. I use standard linear algebra every day.

>Math used to be a young person game, 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 [...] 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

Nonsense.

First, the frontier of mathematics is far closer than you think. Number theory is one of the oldest branches of mathematics, going back thousands of years. There are many more branches that have originated in the latter half of the 20th century. And new ones are coming into existence all the time.

Second, we don't need to focus and specialize — we need to do the opposite. Mathematics is about seeing connections and patterns. We need to teach philosophy and critical thinking, and we need to give people exposure to the vast universe of unexplored — and fun — world of mathematics so that they can go towards the frontiers and push them, rather than spend half a lifetime going in the well-trodden direction.

Finally, undergraduates are still producing new results, every year, in numerous REU programs around the US alone. What gives?

The problem isn't that math is so well-studied that you need so much education to do it. The problem is that we aren't teaching people to do math, we teach them about math that they may or may not use elsewhere.

We don't encourage (or give space) for them to play, experiment, explore, wonder, ask questions, venture into the unknown, and be surprised by what they see (outside the aforementioned REUs).

So many people simply don't get to even start doing mathematics until their third year of graduate school, simply because that's when the structure of our education allows them to.

None of that is necessary. We do it that way because professors are underfunded and mentorship is not rewarded (publish or perish), among other things.

The situation is due to structural problems in academia, not mathematics, humanity, or the advances we made.

Signed, —your neighborhood mathematics PhD

> undergraduates are still producing new results, every year, in numerous REU programs around the US alone.

Speaking as a former undergraduate math student producing “new” results in REU programs, all but a few of these results are completely negligible. There’s still a genius here and there though.

The same can be said about the entire field of mathematics and science in general.

I think it will never get to that. As the frontiers expand, new ways of travel to the frontier are opened up, and the old meandering roads that took so long are left to history.

I haven't spent much time thinking about this but were Einstein's contributions a combination of intellect and "shallow" or relatively quick frontiers. I.e. a very intelligent intellectual working on the problem sets that are approachable without the same level of overhead required these days?

I.e. Today's world wouldn't support an individual who could have such an impact on most math/scientific disciplines?

In a mathematical sense, special relativity is not particularly difficult. Undergraduate physics students often learn it by year two as it doesn't require much more than calculus and linear algebra. That's not at all to downplay Einstein's genius. If it weren't him, I'm sure others would've gotten there eventually, but he was the first there, and his insights were primarily physical (assuming both the constancy of the speed of light in all reference frames and that the laws of physics are the same in all inertial frames of reference).

General relativity was much more difficult. It took Einstein about a decade to develop it and he had to learn differential geometry in order to do so. This work was undoubtedly more profound and required much more advanced mathematics merged with deep insights such as the equivalence principle [0]. This was what made Einstein so successful: the combination of mathematics with physical insights that no one else had put together at the time.

[0] https://en.wikipedia.org/wiki/Equivalence_principle

When it comes to physics, I think the issue is more that we are on the limit of what we can reasonably test. It's like trying to figure out how the body works before the invention of the microscope.

Newton's Gravity was known to be incomplete for many years, just by looking at Mercury. It took several carefully derived experiments studying light for 100s of years that led us to Maxwell's equations in the 1860s that became the basis for Einstein's special relativity in the 1910s. You don't need a billion dollar particle accelerator to come up Maxwell's Equations (2 of them were done by Guass in the 1700s!). 20 or 30 years (and talent!) studying calculus or physics could get you something.

Today we know Einstien's theory of gravity is incomplete but the only places it is incomplete is inside of a black hole (good luck running a test in a blackhole) or at very tiny scales where gravity's affect is minimal. Today, I don't know how you would even come up with a competing theory without having millions to spend on a particle accelerator. While einstein famously just did a thought experiment on what should happen if the speed of light is constant for all observers, most "discoveries" are done now by smashing particles in billion dollar tubes.

> good luck running a test in a blackhole

Theoretically speaking, it's not that hard to run a test inside a black hole (getting to one is the hardest part that we know of). It's communicating the results to anyone else that's "slightly" harder.

Jokes aside, in physics we have both a problem with what can be tested, but also one on the theory/mathematics side. Even for a relatively technical problem, we don't have any good way of solving the actual equations of any slightly complex system in either GR, QM, even Newtonian mechanics. We are actually always relying on numerous approximations and simplifying assumptions, and some of these could themselves lead us astray in some cases.

One day we will be able to download knowledge to our computer-assisted brains, and math will become a young man's game again.

I would add that now we have new tools to enrich mathematicians works: new software tools and computing power. It is not about AI, AI is only a special case or a name for tools that can solve or ASSIST humans in cognitive tasks.

Age is (almost) not a limitation with the right tools, less in the near future.

> we may reach a point where a mathematician's intellectually productive life is not sufficient to contribute Arguably the same could have been said in the 1700s when people lived till a mean life expectancy of about 40 and they would have perhaps been concerned that at some point in their future nobody would be able to make scientific advances in their lifespan. Yet here we are, living till 80s and beyond, making accomplishments in our 50s and 60s.

So we could solve this a few ways:

- Continue to extend the length of human life. Living to 150 will be the norm in a few generations.

- Create AGI that are smarter than us and make the advances for us

- Genetically engineer humans to be twice as smart, and learn as much as an average 30-year-old in 15 years

Or just make mathematical education more efficient which seems like the Occam’s Razor answer to me.

Regarding population decline vs. extreme dexterity... there's surely a good bit of room to improve our ability to discover and foster the dexterous extremes. If "nurture" plays any fraction of a role in the process, there are too many paths through life in this world that would still deny someone the opportunity to achieve their potential dexterity. As a baseline, I'm just talking about abject poverty / famine / war and other horrors that would easily divert someone from even the opportunity to be spotted as a prodigy.

So even if world population declines for a while, I suspect it'd be possible to enjoy more mathematical talent in the coming decades / centuries.

Scott Alexander has an excellent short story on this subject: https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/

This is a big problem in all scientific fields. No one can hold all the relevant information in their head. I expect that there are various "easy" and worthwhile results to be had in many fields if someone could combine and integrate large paths of known results. But this is just not feasible to do for a human mind.

Over time I think we need to either create a GAI or enhance ourselves(via genetic engineering, brain-computer-interface, eternal youth or other ways) to overcome this fundamental human limitations.

While there is a boundless, growing amount of prior work, from (an interested) outsiders point of view there also seems to be progress on discoverability and formalisations around how this is expressed. Building on the abstractions of others seems like it would provide an environment for _more_ novel work.

The abstractions need to be understood, but potentially only to a level of n-1. Isn't this one of the core principles of proofs?

I can't pretend to be anywhere near the frontiers of math but aren't some contributions about providing a unifying framework or simplifying re-expression of something that was already known in a way that makes it more comprehensible? Is there not always hope that the right new perspective or innovation in one generation will allow subsequent generations to get up to speed faster?

Point is that the low hanging fruits are finite and are picker continuously, thus sooner or later they run out. Reinventing something I consider low hanging just not recognized for some time.

It's not really plausible that such a process could continue forever. It may be plausible that there are some better foundations to various fields of math that could reduce the size of the proof for Fermat's last theorem by, say, 100 fold - but it's not plausible that it could ever be fitted onto the side of a page with the right abstraction. It's not plausible that, even in the far future, you'll be able to learn the information needed to derive formulas you'd learn in post-graduate maths today after a 1 week course in 5th grade.

On the other hand, it's extremely plausible that math as an abstract idea could build arbitrarily complex structures that would be of interest if we were able to learn to model them - so, almost necessarily, within the limitations of current human cognition and/or lifespan, there will be some structures that it is impossible for a human to learn.

Of course, in some arbitrary future we may find that those limitation of human cognitition/life-span can themselves be overcome - if that's the case, then the argument changes significantly.

This effect clearly exists. It suffices to check how physics and math were expressed in the papers that discovered them and how they are currently represented to notice that sometimes organizing knowledge in the right framework and simple notational changes can be sufficient to power up our ability to make new advances.

Question is, are we dedicating enough effort to this endeavors? Is rewriting known math under a different guise sufficient to survive the publish or perish attrition? And will these efforts keep returning results?

I suspect the answer is no to the first two questions, and I don't know the answer to the third.

Wouldn't Scholze be a counterexample to the thesis that young people can't do fundamental contributions?

I would say that it supports the idea that only young people are able to change the fundations of maths, while older people can still use the existing techniques to push the bord, specially on those super technical fields where experience and knowledge are an advantage.

Maths always used to be an entertainment for aristocrats and philosophers.

Only in very few exceptional times in history it was been used for something else, being the largest exceptions the industrial and scientific revolution, and these usually did not require breakthroughs: just apply existing knowledge.

It’s never been a young person game.

This doesn’t account for growing literacy and numeracy in the developing world, we might be nearing peak population but we are still ages away from a globally educated population with both the resources and education to start contributing to academia/research.

Similar situation with energy generation. Low hanging fruit is pretty much gone. Current output relies heavily on advancing up a tech ladder. If we somehow have to start over it might not be possible to make it back.

We detached this subthread from https://news.ycombinator.com/item?id=33513127.

We will have super advanced AGI exploring the frontiers of Math and Physics. Check out the recent Lex Fridman pod with Andrej Karpathy. Fascinating discussion.

Definitely a potentially meaningful reason as to why we should invest more heavily into life extension.

I don’t foresee AI overtaking human cognition for at least a decade or two.

Not nit picking, just informing, population currently stated to peak around 2080-2100 at around 10.2-11.4b, depending on the estimate!

You forget, we are becoming cyborgs. Computers will allow us to move much faster, work as a team, and see the future

Perhaps math focused hot-housing at a younger age could provide an extension?

(genuine question, as I don't know anything about it)

> Perhaps we could then rely on computer assisted theorem provers.

Or AI, particularly if we figure out AGI.

Or we innovate elsewhere.

On a similar theme, Scott Alexander's short story Ars Longa, Vita Brevis https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/

Analogously, I've heard transhumanists make the argument that as fluid intelligence declines as crystallized intelligence grows, we have yet to see a human mind in its full power.

There's a great SSC story about this concept: https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/

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