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This is essentially a critique of specialization, applied to the field of mathematics. I think that this trend might be inevitable, not just in the field of mat

by fh 17y ago

This is essentially a critique of specialization, applied to the field of mathematics. I think that this trend might be inevitable, not just in the field of mathematics. The total amount of knowledge that humanity has accumulated continues to grow. If the amount of science that a single person can understand is finite, then individual scientists will understand an ever smaller percentage of it. In some ways that makes me sad.

thras 17y ago
Maybe that's the reason. Or is it that the modern academic job scene pushes people to be big fish in small ponds rather than medium-sized fish in bigger ponds?
gregwebs 17y ago
I think a critique that could try to control against expanded knowledge/specialization would be to try to look at the big problems in the field. Is specialization helping people tackle them with their deep knowledge or avoid them to work on things that are irrelevant?
andreyf 17y ago
This trend is inevitable if all of the "specialized" branches and sub-branches really have independent meaning. It's hard to talk concretely about this, but my intuition says this is highly unlikely: chances are that connections between seemingly unrelated subfields should allow us to unify concepts and discover more meaningful mathematical truths.
baddox 17y ago
I don't see why two specialists couldn't link their specific knowledge when necessary by explaining it in the context of the fundamental knowledge they should all share. That is to say, as a person specializes more and more in mathematics, there's no reason to think they should forget the earlier lessons they've learned.
- msluyter 17y ago
  I visualize the situation like a search tree. As you travel down the tree away from the root, you become increasingly isolated from other branches of the tree, and you can't even talk to distant leaf nodes because you lack the same vocabulary and shared set of theorems. Yes, we share some parent way up the tree, but we can't go all the way back to that as a starting point because it'd simply take too long to progress from there back down to the interesting leaf node. This vision makes me wonder when mathematics will reach a point where mastering the material required to understand a leaf node will take greater than the average life span.
  - loup-vaillant 17y ago
    > I visualize the situation like a search tree. I disagree. I observed that, until up to graduation, the situation is like a search graph. When I studied, I noticed many interdisciplinary connections. Fields like mechanic, electronic, logic, automatic, programming, often look like two sides of the same coin. However I also noticed an inability from others to see the damn connections. If you present the same thing from another angle, they often fail to see it as the same thing, while I factor the obvious pattern like you would code (I've seen it in the case of 3D vision). The consequences are quite catastrophic: the languages (jargon) used to describe each discipline diverge, making it even more difficult to notice the similarities. This convince even more people that there isn't any connection. That the situation is like a tree. At this point, they don't even bother to seek the connections, and we end up with a disconnected mess. > This vision makes me wonder when mathematics will reach a point where mastering the material required to understand a leaf node will take greater than the average life span. That can't happen: more time to study relevant material means less time to push further. It will take many geniuses to be able to learn and push fast enough. But this is pointless. If it takes you a lifetime to learn a particular field, that field will simply fall into oblivion. So, the geniuses (at least the one worth mentioning) won't push further. They will simplify their field, reducing the time required to learn it.

by fh 17y ago

thras 17y ago

Maybe that's the reason. Or is it that the modern academic job scene pushes people to be big fish in small ponds rather than medium-sized fish in bigger ponds?

gregwebs 17y ago

I think a critique that could try to control against expanded knowledge/specialization would be to try to look at the big problems in the field. Is specialization helping people tackle them with their deep knowledge or avoid them to work on things that are irrelevant?

andreyf 17y ago

This trend is inevitable if all of the "specialized" branches and sub-branches really have independent meaning. It's hard to talk concretely about this, but my intuition says this is highly unlikely: chances are that connections between seemingly unrelated subfields should allow us to unify concepts and discover more meaningful mathematical truths.

baddox 17y ago

I don't see why two specialists couldn't link their specific knowledge when necessary by explaining it in the context of the fundamental knowledge they should all share. That is to say, as a person specializes more and more in mathematics, there's no reason to think they should forget the earlier lessons they've learned.

msluyter 17y ago
I visualize the situation like a search tree. As you travel down the tree away from the root, you become increasingly isolated from other branches of the tree, and you can't even talk to distant leaf nodes because you lack the same vocabulary and shared set of theorems. Yes, we share some parent way up the tree, but we can't go all the way back to that as a starting point because it'd simply take too long to progress from there back down to the interesting leaf node. This vision makes me wonder when mathematics will reach a point where mastering the material required to understand a leaf node will take greater than the average life span.
- loup-vaillant 17y ago
  > I visualize the situation like a search tree. I disagree. I observed that, until up to graduation, the situation is like a search graph. When I studied, I noticed many interdisciplinary connections. Fields like mechanic, electronic, logic, automatic, programming, often look like two sides of the same coin. However I also noticed an inability from others to see the damn connections. If you present the same thing from another angle, they often fail to see it as the same thing, while I factor the obvious pattern like you would code (I've seen it in the case of 3D vision). The consequences are quite catastrophic: the languages (jargon) used to describe each discipline diverge, making it even more difficult to notice the similarities. This convince even more people that there isn't any connection. That the situation is like a tree. At this point, they don't even bother to seek the connections, and we end up with a disconnected mess. > This vision makes me wonder when mathematics will reach a point where mastering the material required to understand a leaf node will take greater than the average life span. That can't happen: more time to study relevant material means less time to push further. It will take many geniuses to be able to learn and push fast enough. But this is pointless. If it takes you a lifetime to learn a particular field, that field will simply fall into oblivion. So, the geniuses (at least the one worth mentioning) won't push further. They will simplify their field, reducing the time required to learn it.