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I was thinking about this today and it seems like a good question. Assuming mathematics will keep on expanding, do you think it will ever become impossible for a beginner to learn all the known material on a subject (such as mechanics), simply because there is so much prerequisite material to learn? Of course, this will have to be in the very far future, because at the moment we can still get talented 15 year olds up to speed due to the compressed material available in textbooks, which contains centuries worth of research in a few hundred pages.

Hope you understood my question, any replies are appreciated.

Thanks.

David Richerby
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Ninja Boy
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    It seems to me that mathematics is expanding in breadth more than in depths these days (at least this is definitely happening in algebra). Knowing all of a subject will become harder but knowing enough to do new research will not. – darij grinberg Nov 23 '14 at 23:20
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    It's been impossible for a beginner to learn all the known material on a mathematical subject since the late 19th century. While that causes various problems, the mathematical community has adapted to it and keeps producing new mathematicians. – Kevin Carlson Nov 23 '14 at 23:20
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    This must be a question that everyone has thought about at some point during their education. – Auberon Nov 23 '14 at 23:23
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    As a mathematician-in-training I would say it's a little intimidating, knowing how much as already been done, but at the same time that goes for anyone my age in any field today. Mathematics is often said by non-mathematicians to be among the slowest changing of all fields related to science and mathematics -- now I know that in some sense to be completely untrue. In response to your question, I would say the only thing someone like me can do is worry less about broad subjects and attack finer and finer details in smaller, but still significant, problems. – bjd2385 Nov 23 '14 at 23:26
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    Not so many years ago - perhaps 250 -- one might have asked "Can one really understand all of natural philosophy?", which included mathematics, botany, physics, ... Since then, we've made narrower domains, and those who study physics are no longer ashamed not to know botany. Fields will divide until they're small enough for those who study them to know ...well, to know enough. – John Hughes Nov 23 '14 at 23:38
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    I think we already try to cram far too much into the heads of bright teenagers via those "compressed" textbooks. In other words, there's already far too much math one is expected to learn even at "entry" level for students who will go into mathematical fields (such as math itself, the "hard" sciences, or engineering). – David K Nov 24 '14 at 02:26
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    I love this meta-math question that involves math. Since we're in the information age, we can easily access loads of info so our short-term memory can retain and calculate a lot. Maybe if bioengineering will some how improve our brains, then this may not be an issue? – But I'm Not A Wrapper Class Nov 24 '14 at 04:00
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    Consider Poincare, who many call the last universalist. With him died the hope of learning all of mathematics. – user795305 Nov 24 '14 at 04:41
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    Please see this illustrated guide to the nature of in-depth research and the acquisition of knowledge required to support it. Research and knowledge are merely reflective of the greater truth that this one fleeting lifetime can only ever cover an infinitesimal sliver of the whole of possible experience. Make the best of it. – Dan Bryant Nov 24 '14 at 14:50
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    Another big way to keep growing is by appropriately compressing information. Often the first proofs of a theorem are only accessible to the experts. Then the most important ideas are distilled and disseminated, so the proof can be rewritten in a way making it comprehensible to most people in the subfield. A couple decades down the line and it is in undergraduate books. "Gains" in exposition and understanding of what "really" makes something tick help get the next generation of scholars up to speed faster. – Steven Gubkin Nov 24 '14 at 19:21
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    You can always just keep making the specialties narrower and narrower, but, practically speaking, the ability for one person to "understand math" was probably lost 100 years ago. The problem is not simply memorizing material, but truly understanding the concepts of geometry, calculus, game theory, cryptology, probability, chaos theory, field theory, quantum mechanics, and a hundred others. It is beyond the abilities of any mere mortal. – Daniel R Hicks Nov 24 '14 at 20:49
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    a subquestion of that is: is there a point where to discover something new, one must spend more that a lifetime learning what is already known? (even in a very narrow field) – njzk2 Nov 24 '14 at 21:45
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    In future, humans might not have to learn in the way we do. All necessary data would be fed directly into the brain. We are already making progress in brain to brain communication: http://nrecursions.blogspot.in/2014/08/brain-to-brain-communication.html Alternatively, our brains could get wired to a central 'cloud' of data to access information the same way we currently access the internet. – Nav Nov 25 '14 at 03:10
  • If to do so we must re-think our way of teaching math. – Karl Morrison Nov 25 '14 at 16:14
  • It is already impossible to learn all of math, and always has been. That has nothing to do with the possibility of learning math, any more than it's impossible to learn to read just because you can't read everything. – Jim Balter Mar 07 '15 at 00:22
  • I think you may be right, in the future it may become impossible for a person with human intellect to comprehend some branch of math in detail in only 5 years. But by that time hopefully people will live much longer, and perhaps genetic engineering will allow people to be much smarter as well. – Ovi Aug 16 '17 at 05:22

8 Answers8

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I think this question is quite analogous to this one:

Will it become impossible to learn music?

One can argue quite convincingly that some branches of music have a deep theory, and to master it (in a certain technical sense) is almost impossible. But the point is: music isn't about mastering an aspect. It's really about interconnecting experiences in order to add, increase and even modify the existing "theories". In mathematics, where the knowledge builds up on itself most explicitly when compared to some other fields, it is common to think that it is a characteristic of mathematics itself: the neverending things to learn about something; there is always something more about the subject. But this isn't exclusive to mathematics at all. Everything always has something else. So, in some sense, the answer to your question is yes.

... But humanity always finds something new. New concepts, new ideas, new ways to think about something. This is where the analogy to music kicks in: some people, for example, may have thought that classical music closed it: we have all we need. Then came jazz. Some professors may have thought that physics was closed: we have all we need...

"in this field, almost everything is already discovered, and all that remains is to fill a few holes." - Quote from a professor to Max Planck, advising him not to study physics. Planck was one of the prominent names of quantum physics.

We are humans. We will always come up with some new crazy stuff. This answer your question as a no: because we will always create new things to learn.

Aloizio Macedo
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This was one of the plot lines in sci-fi novel Incandescence by Greg Egan.

It's set in the far future when pretty much all of life's problems have been solved, including knowing most of mathematics. Since, in the book, death is no longer a problem, there is a group of characters who've been spending 1600 years trying to prove a new theorem.

So it's perfectly easy to envisage a scenario where being able to do new mathematical research becomes onerously difficult. Of course, that time will be farther away than you think, due to new cutting edge research being converted into easier to understand knowledge (via textbooks, etc.), and research tools improving (where's my AI that can solve theorems already?), and people living longer to have more time to get up to speed.

Richie Cotton
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    I don't know this book, but I am intrigued. I think this may shed some light on the original question: there could be an upper limit on the depth of possible research imposed by the human lifespan. – Michael Lugo Nov 24 '14 at 15:10
  • Hmm, in the eyes of immortals, spending 1600 years on a math problem might be like spending 16 weeks for us: a pretty long project, but not unacceptable. – MaudPieTheRocktorate Jun 19 '18 at 01:16
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This might not be an adequate answer as it's subjective in nature, but from a personal perspective, I don't think it's ever possible to learn the whole of a topic completely. Visiting this site once or twice a week shows the variety of different approaches to any different problem. But it's fun to try!

Mike Miller
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It's already impossible to learn all of any area of mathematics and has been for at least a hundred years (probably 150+). If you drill down into the detail, you'll find that the "world expert in X" is actually somebody who is very knowledgeable about X and, particularly, the sub-field X.Y and maybe, just maybe, knows everything that is known today about the specific area X.Y.Z. But even that can't be quite true since, if any other person in the world is working on X.Y.Z, then that person knows all the stuff they invented this week and haven't had chance to write a paper about and, unless he's telepathic, our world expert doesn't know those things.

But that doesn't mean that "learning mathematics" becomes impossible. The maths we learn today in schools and universities will still be perfectly valid in a thousand years' time so, as an absolute worst-case, you could still teach a math class in 3014 by pretending that nothing new had been learnt in the preceding millenium. The only real problem with that approach is that our selection of topics in 2014 might not be relevant to 3014. For example, there's a tendency to include more discrete mathematics in school syllabi at the moment, presumably because of its applicability to computer science. Who knows, maybe in a thousand years' time, there'll be some killer application of group theory that means we teach much more of that and much less calculus at high-school level.

do you think it will ever become impossible for a beginner to learn all the known material on a subject (such as mechanics), simply because there is so much prerequisite material to learn?

If you look closely at this question, you'll see that it already contains its own answer. It's already impossible to learn all of mechanics, precisely because there's so much of it and there are so many prerequisites. This means we'll never run out of basic mathematics to teach schoolchildren. If you want to teach kids wormhole hypermechanics but you can't because of all the background they'd need, you teach them that background, instead, and use the wormholes to motivate it.

David Richerby
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For example you can learn $F=ma$ in college. You may able to solve all the text book problems about it. But when you look at this formula each time you will figure out that you still didn't understand everything about it.

In mathematics you don't understand things. You just get used to them.

John von Neumann

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newzad
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First, let us assume homo-sapiens won't destroy all form of civilization in the near future.

With current rate of technological improvement, it is highly likely before the end of this century, we will have a working direct brain-computer interface.

Human's current limitation of understanding, memory will be lifted and our brain's processing power will be significantly enhanced. It is simply too early to predict we as a speices will become impossible to learn math.

In the worst case, machine can rise and replace our species on the tower of intelligence.

achille hui
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In any field, the knowledge fields expands fast. Fast enough to make it impossible to know everything. In fact the more you study a particular field (maths, electronics, mechanics etc...) the more you feel there is a lot to learn, and the more you feel you don't know much. That is why being a good team player is always a requirement in any job, because skills that different people have complement, and the it's the skills of each individual added together that allows the team to cope with new knowledge and build something.

harvey
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I have wondered if in the future we will have organic vats dedicated to AI. Will we be able to give one of these vats some data and assumptions. Leave them be for a few years and see what comes out of it. Will they be so far outside the box that we will not even be able to comprehend how they came up with what they did. This is a bit of a science fiction fantasy.

I do believe that learning and understanding will some day be beyond us. Our future challenges will be understanding how to apply what has been thought through by our creations.

john
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