21 ms·

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 u

• tsol 4y ago
  >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.
  • Frost1x 4y ago
    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.
    • novalis78 4y ago
      This is where longevity research a la Harold Katcher comes in and allows for super centenarian geniuses to shape the future of humanity.
      • nerdbert 4y ago
        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.
        • novalis78 4y ago
          Harold Katcher: “The Illusion of Knowledge”.
      • pbronez 4y ago
        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.
    • lupire 4y ago
      Even with optimal compression, there is still a finite minimum size.
  • chairhairair 4y ago
    Here is a short story that covers that idea: https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/ https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/
    • philipswood 4y ago
      Thanks for posting it! I've read it, lost it, and have been searching for it for a few years now.
  • comboy 4y ago
    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.
  • snovv_crash 4y ago
    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.
    • tsimionescu 4y ago
      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.
      • snovv_crash 4y ago
        > 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.
• JasonFruit 4y ago
  Or we innovate elsewhere.
• gavagai691 4y ago
  "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."
  • danuker 4y ago
    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.
    • skyde 4y ago
      do you have experience with CAS? i would love to learn how to use CAS to write proof more effectively
      • rowanG077 4y ago
        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.
      • dagw 4y ago
        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.
• elefanten 4y ago
  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.
• rtpg 4y ago
  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)
  • rnvannatta 4y ago
    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.
    • BlueTemplar 4y ago
      I am somewhat unconvinced that quaternions are more challenging, or at least a worse way to think about the issue : https://eater.net/quaternions https://eater.net/quaternions And speaking of Maxwell's Equation"s" : http://www.av8n.com/physics/maxwell-ga.htm#sec-preview http://www.av8n.com/physics/maxwell-ga.htm#sec-preview
      • rnvannatta 4y ago
        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.
• kimburgess 4y ago
  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?
• boringg 4y ago
  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?
  • vecter 4y ago
    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 https://en.wikipedia.org/wiki/Equivalence_principle
  • nemothekid 4y ago
    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.
• __MatrixMan__ 4y ago
  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.
  • deleted 4y ago
    [deleted]
  • nephanth 4y ago
    Homotopy type theory?
  • sdenton4 4y ago
    My favorite thought game along these lines is imagining an alien civilization where permutations are the fundamental object instead of the natural numbers.
    • __MatrixMan__ 4y ago
      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.
  • Qem 4y ago
    This approach reminds what the geometric algebra folks are trying to do.
• mgfist 4y ago
  One day we will be able to download knowledge to our computer-assisted brains, and math will become a young man's game again.
• Armic 4y ago
  There's a great SSC story about this concept: https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/ https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/
• maegul 4y ago
  > 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.
  • trash3 4y ago
    This sentiment is humorous, as I'm an optimist apparently. More children will learn more quickly. Applying AI to education should supercharge the smartest.
    • deanCommie 4y ago
      > Applying AI to education should supercharge the smartest. What does that mean?
      • wnkrshm 4y ago
        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.
      • kwhitefoot 4y ago
        As we do not yet have artificial intelligence making confident statements about what its effects might be is surely rather rash.
    • ConceptJunkie 4y ago
      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.
• firecall 4y ago
  Perhaps math focused hot-housing at a younger age could provide an extension? (genuine question, as I don't know anything about it)
• pie_flavor 4y ago
  Scott Alexander has an excellent short story on this subject: https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/ https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/
• rowanG077 4y ago
  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.
• nonethewiser 4y ago
  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.
• notacop31337 4y ago
  Not nit picking, just informing, population currently stated to peak around 2080-2100 at around 10.2-11.4b, depending on the estimate!
• phlakaton 4y ago
  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.
• lawrenceyan 4y ago
  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.
• wheels 4y ago
  > [...] 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.
  • data_maan 4y ago
    > 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).
    • matkoniecz 4y ago
      > 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 https://en.wikipedia.org/wiki/Cultural_Revolution#Death_toll alone
  • germandiago 4y ago
    A self-made man :)
• SkyMarshal 4y ago
  > Perhaps we could then rely on computer assisted theorem provers. Or AI, particularly if we figure out AGI.
• abeppu 4y ago
  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?
  • szundi 4y ago
    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.
  • tsimionescu 4y ago
    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.
  • pa7x1 4y ago
    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.
• nudpiedo 4y ago
  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.
• FeepingCreature 4y ago
  On a similar theme, Scott Alexander's short story Ars Longa, Vita Brevis https://slatestarcodex.com/2017/11/09/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.
• dheera 4y ago
  > 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
  • rcarr 4y ago
    Or just make mathematical education more efficient which seems like the Occam’s Razor answer to me.
• romwell 4y ago
  >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
  • oefrha 4y ago
    > 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.
• wslh 4y ago
  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.
• kingkongjaffa 4y ago
  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.
• alexnewman 4y ago
  You forget, we are becoming cyborgs. Computers will allow us to move much faster, work as a team, and see the future
  • deleted 4y ago
    [deleted]
• coutego 4y ago
  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.
• deepzn 4y ago
  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.
• dang 4y ago
  We detached this subthread from https://news.ycombinator.com/item?id=33513127 https://news.ycombinator.com/item?id=33513127.