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"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 anythin

"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.
  - danuker 4y ago
    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.
sillysaurusx 4y ago
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/1667865208.JPG 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.
- mlyle 4y ago
  Physics is limited by having to represent phenomena in our physical world simply. Mathematics is not just a small integer multiple larger than this.
  - joe_the_user 4y ago
    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.
    - romwell 4y ago
      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.
- nyokodo 4y ago
  > 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?
- squokko 4y ago
  But you can manufacture new areas of mathematics. For example, Conway's Game of Life, and then prove theorems on it.
lupire 4y ago
> 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.
- ves 4y ago
  Most of the now-deep, now-interesting areas were once shallow and uninteresting.
carrolldunham 4y ago
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
- balsam 4y ago
  Or, as Juergen Schmidhuber likes to point out, in the same field.
sdenton4 4y ago
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...
pa7x1 4y ago
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.

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.
- danuker 4y ago
  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.

sillysaurusx 4y ago

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/1667865208.JPG 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.

mlyle 4y ago
Physics is limited by having to represent phenomena in our physical world simply. Mathematics is not just a small integer multiple larger than this.
- joe_the_user 4y ago
  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.
  - romwell 4y ago
    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.
nyokodo 4y ago
> 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?
squokko 4y ago
But you can manufacture new areas of mathematics. For example, Conway's Game of Life, and then prove theorems on it.

lupire 4y ago

> 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.

ves 4y ago
Most of the now-deep, now-interesting areas were once shallow and uninteresting.

carrolldunham 4y ago

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

balsam 4y ago
Or, as Juergen Schmidhuber likes to point out, in the same field.

sdenton4 4y ago

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...

pa7x1 4y ago

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.