Specialization is something I often wonder about. Consider the following analogy: the sum of human knowledge is represented by a giant circle (though not necessarily perfectly circular in shape). Anything inside the circle is that which is known; anything outside is unknown. You're born in the center, and as you learn, the set of what you know extends towards the edge. As you mentioned, at some point in time, you could know all of the circle; now you can only know some subset of it. If you're a scientist or whatnot, you actively move the edge outward.
My question is, will we ever reach a point where it takes so long to move from the center to the edge that we'll eventually stop making progress? In mathematics, it already takes considerable time to reach the knowledge boundary (30 years or so?) Will we reach a point where it takes greater than a lifetime? Or, is it possible to jump to the edge by skipping some foundational knowledge (could you do higher math without knowing, say, calculus, for example?) What are the risks of doing so? And ultimately, how does a complex society hang together when each person has (at most) knowledge of a tiny subset of available knowledge?
I'd argue that, as advances are made, it becomes easier to move faster through the circle. It's not that you're skipping foundation knowledge, but you're just learning one core piece. Instead of learning the area under an encyclopedia of curves, you learn integral calculus and you're done. Instead of memorizing thousands of rules involving the interaction of various charged objects, you learn Maxwell's four equations and you're done. Or, you could learn geometric calculus and then only need to learn one four-symbol equation. Instead of learning thousands of different circuits to accomplish thousands of different tasks, you learn one Turing machine.
Knowledge is constantly increasing, but we're also continually finding that a large amount of our previous knowledge was redundant. I once saw a great article, that I wish I could find again, list off a series of scientific laws that could all be derived from nothing but unit analysis. The circle grow larger, but our steps do as well.
Good point. There are algorithms for manipulating Roman numerals that used to be taught, but that no longer are. And really, little is lost.
People used to practice at long algebraic exercises (e.g., trig manipulations, or tricky integrals) that are no longer considered worthwhile. Just give it to Mathematica. It's like medium-sized linear systems -- sure, you could solve it manually, but nobody does, or even remembers all the shortcuts and tricks that used to be so important.
Maxwell's equations aren't enough to solve any practical example systems in the real world, so they aren't all you need to know to understand the world.
It's often been said that Gottfried Wilhelm Leibniz was one of the last people to master and contribute to all of the fields of knowledge in existence during his time (eg http://plato.stanford.edu/entries/leibniz/ ).
Speaking of just mathematics, we're WELL past the point where it's possible to cover the entire interior of the "mathematical circle". As an illustration, MathSciNet (http://www.ams.org/mathscinet/) is a mathematical review database that consists of prettymuch every mathematical book and peer-reviewed article in mathematics since the 1940's (and some well before 1940). As of this writing, there are just under three million publications stored in MathSciNet. And these articles are so specialized that a researcher in one area would have to exert a significant amount of effort to understand articles in a completely different area of research.
So, how do we go from the center of the mathematical circle to the edge? By starting with the basics and moving through a path of what's considered "important" (ie the broad survey provided by undergraduate mathematics courses: the calculus sequence, real analysis, abstract algebra, combinatorics, topology, etc) along the way.
You get at the degree to which all human knowledge is limited by the central nervous system that humans possess. There must be some finite number of axons and dendrites that can be stuffed into the cranium of a human, and even if there is a sophisticated system of compression and/or multiplexing in use, we can probably assume that a finite amount of axons and dendrites can only carry a finite amount of information.
Thinking of the physical limits on human knowledge leads to a model of knowing that puts the central nervous system at the center of our our experience of the universe. There may be an absolute reality that exists outside of us, but since all of us are human (I assume) we must admit that our knowledge of that absolute reality will forever be shaped by that central nervous system.
The limits of the central nervous system cause humans to look for models of reality that fit well with those limits. For instance, there is The Magical Number Seven, Plus or Minus Two:
Because of the (very very finite) limits on working memory, the human mind leans heavily on heuristics that allow simple models to stand in for the complexity of reality. No human being could easily work with a physics model that had 1 million variables: therefore we tend to look for models that have a handful of variables, yet still give us reasonably good estimates.
Just yesterday there was posted to Hacker News an article about a slinky, and that article contains a good demonstration of the kind of simplifying assumptions we often make to reason about problems that are too complex to approach directly:
"I decided to idealize the problem like this: the slinky is an ideal spring with mass distributed uniformly throughout. It is also a spring that can pass through itself. These assumptions make analyzing the problem easier."
And of course, we often present students with problems where the difficulty of achieving perfect measurements of variables is simply assumed away. For instance in this exercise:
The important things to measure are: force, deflection, torque, moment of inertia, stress and strain. Often, in real life situations, engineers have no way to perfectly measure these things. Large engineering projects proceed with various methods for getting reasonable estimates: known upper and lower bounds, prototypes, isolated testing of particular connections, etc.
The smartest people alive never get to the point where they really know reality, they only know estimates. I do not mean to make this point overly philosophical, I am talking about practicalities here: there may be an absolute reality that we can know directly, but when it comes to doing anything useful, we develop estimates using fairly simple models, models which are well suited to the limits of our minds.
In some sense, the models we build about the world are always simplified models that make it easier for our limited central nervous systems to make estimates about the world.
I just offered a physics examples of simplifying assumptions, but I could also focus on the humanities. Perhaps your life's goal is to figure out the "real" reasons why the French Revolution happened, or why Communism was so popular for so long, or why the Industrial Revolution happened -- the possibilities are as limitless as the combination of all the variables in action over the course of centuries, the psychology and biology of billions of people, and the technology that existed at the time, and the weather and the crops and the economy and the religions and the amount of sunlight reaching the earth and everything else, combining exponentially to some very large sum. You might research the issue for 40 years, but in the end, if you want to draw some useful conclusion, something that others can benefit from, something that might influence policymakers in the future, you will need to come up with a simplified model that focuses on the handful of variables that you regard as the most important.
There is another model of knowing, which takes for granted the absolute reality outside of us. That model is well summarized by this XKCD comic:
In that model, biology is a sub-branch of physics, which is a sub-branch of math. However, if you focus on the limits of the central nervous system as the practical limit of human knowing, then you end up with a model where physics and math (and everything else) is a sub-branch of biology, specifically, a sub-branch of neurology. Even the two models of human knowing that I describe in this post to Hacker News can be thought of as sub-branches of neurology. All human knowing is, in some sense, a clever hack of our neurology, and certainly all heuristics and models are hacks which adapt to the limits of that neurology.
There is nothing wrong with simple models, so long as they allow us to get useful work done. But I think our reliance on simple models tells us why specialization is so important. If a person can study a subject for 40 years and, having read tens of thousands of documents, still distills the data down to a small handful of variables, a model that gracefully adapts to the limits of the human mind, then we must admit that it is the rare human being who goes much beyond simple models. In those cases where simple models don't work, and where there only thing that works is a model with hundreds or thousands of variables, then clearly, achieving an understanding of such models must surely be the work of a lifetime. You need to get to studying when you are 20, and you need to specialize, because when you are 50 you will still be only approaching the ideal, never quite there, only approaching it, still learning more.
You get at the degree to which all human knowledge is limited by the central nervous system that humans possess.
I think that our current revolution, the Internet and computers in general, is allowing us to overcome these limits.
The question is how to integrate them better, how to improve the link between humans and computers, and how to improve our own cognition in the most natural, most human way to get the most out of the whole of human knowledge that is now available everywhere.
Don't forget writing. Before writing (in some form or another), humans just had to remember things. Writing allowed them to recall things over a much greater period of time, as well as share them.
I don't think the question is about how much we can store in our brains, but rather how much we can cram in there within a lifetime. I think we'll always reach death before we fill our brains. Of course, if we ever reach a true singularity, transfer our consciousnesses into robots, and live forever, the one question becomes the other.
And are there natural boundaries and are 3-d space conventional area vs volume concepts relevant if whole new areas open up?
For example there was a relevant fear at one point that if we continued to discover another new chemical element every two decades or so, in a couple centuries no one could possibly know all the elements, perhaps not even name them all. But that growth had a natural border in the triple digit range; no problemo. Of course chemists found entirely new things to spend time on, rather than just discovering new elements.
There may very well be nothing more that can be learned WRT classical mechanics. Plenty of fun quantum and relativistic stuff to learn, but thats not on the "here be dragons" map which is now completely explored for classical mechanics, or as near to completely explored not to matter. Someday those fields will be fully explored and boring and closed to further exploration.
Another interesting topic is eventually stuff is forgotten when its irrelevant. What happens in a million years when most academic activity is reinventing useless stuff thats already been invented, found useless, and forgotten, so how do you stop the backlash from eliminating actually useful knowledge?
Well, a million years is about 5 times the lifetime of our species. Whoever will be reinventing our knowledge will probably be so different from us that this discussion won't make any sense anymore.
A few thousand years ago, somebody probably asked "How will people of the future be able to tell histories once we paint over all the rocks?"
"Whoever will be reinventing our knowledge will probably be so different from us that this discussion won't make any sense anymore."
You're going to make contemporary archeologists cry, saying things like that. And the future ones are likely to be even better at it.
That might make an interesting very long term startup idea, in a million years it may make a heck of a lot more sense to hire archeologists to "discover" new areas for business profit than to hire primary creators to try and re-invent the hard way, the same way the first 345 times "it" was invented. And I'm only slightly tongue in cheek that this is already how contemporary IT works.
Your other comments remind me of what Lagrange said in 1781 about mathematics:
"...I am not sure that I shall still do geometry ten years from now. I also think that the mine is already almost too deep, and must sooner or later be abandoned. Today, Physics and Chemistry offer more brilliant discoveries and which are easier to exploit"
Comments
Specialization is something I often wonder about. Consider the following analogy: the sum of human knowledge is represented by a giant circle (though not necessarily perfectly circular in shape). Anything inside the circle is that which is known; anything outside is unknown. You're born in the center, and as you learn, the set of what you know extends towards the edge. As you mentioned, at some point in time, you could know all of the circle; now you can only know some subset of it. If you're a scientist or whatnot, you actively move the edge outward.
My question is, will we ever reach a point where it takes so long to move from the center to the edge that we'll eventually stop making progress? In mathematics, it already takes considerable time to reach the knowledge boundary (30 years or so?) Will we reach a point where it takes greater than a lifetime? Or, is it possible to jump to the edge by skipping some foundational knowledge (could you do higher math without knowing, say, calculus, for example?) What are the risks of doing so? And ultimately, how does a complex society hang together when each person has (at most) knowledge of a tiny subset of available knowledge?
I'd argue that, as advances are made, it becomes easier to move faster through the circle. It's not that you're skipping foundation knowledge, but you're just learning one core piece. Instead of learning the area under an encyclopedia of curves, you learn integral calculus and you're done. Instead of memorizing thousands of rules involving the interaction of various charged objects, you learn Maxwell's four equations and you're done. Or, you could learn geometric calculus and then only need to learn one four-symbol equation. Instead of learning thousands of different circuits to accomplish thousands of different tasks, you learn one Turing machine.
Knowledge is constantly increasing, but we're also continually finding that a large amount of our previous knowledge was redundant. I once saw a great article, that I wish I could find again, list off a series of scientific laws that could all be derived from nothing but unit analysis. The circle grow larger, but our steps do as well.
Good point. There are algorithms for manipulating Roman numerals that used to be taught, but that no longer are. And really, little is lost.
People used to practice at long algebraic exercises (e.g., trig manipulations, or tricky integrals) that are no longer considered worthwhile. Just give it to Mathematica. It's like medium-sized linear systems -- sure, you could solve it manually, but nobody does, or even remembers all the shortcuts and tricks that used to be so important.
A brief and vivid essay on this is "Forget it" by the late Isaac Asimov: http://readanybooks.net/ScienceFiction/Asimov37/27283.html
That's reductionist and oversimplifying.
Maxwell's equations aren't enough to solve any practical example systems in the real world, so they aren't all you need to know to understand the world.
The circle-knowledge analogy is depicted succinctly in Matt Might's Illustrated guide to a Ph.D. ( http://matt.might.net/articles/phd-school-in-pictures/ ).
It's often been said that Gottfried Wilhelm Leibniz was one of the last people to master and contribute to all of the fields of knowledge in existence during his time (eg http://plato.stanford.edu/entries/leibniz/ ).
Speaking of just mathematics, we're WELL past the point where it's possible to cover the entire interior of the "mathematical circle". As an illustration, MathSciNet (http://www.ams.org/mathscinet/) is a mathematical review database that consists of prettymuch every mathematical book and peer-reviewed article in mathematics since the 1940's (and some well before 1940). As of this writing, there are just under three million publications stored in MathSciNet. And these articles are so specialized that a researcher in one area would have to exert a significant amount of effort to understand articles in a completely different area of research.
So, how do we go from the center of the mathematical circle to the edge? By starting with the basics and moving through a path of what's considered "important" (ie the broad survey provided by undergraduate mathematics courses: the calculus sequence, real analysis, abstract algebra, combinatorics, topology, etc) along the way.
Illustrated: http://matt.might.net/articles/phd-school-in-pictures/
You get at the degree to which all human knowledge is limited by the central nervous system that humans possess. There must be some finite number of axons and dendrites that can be stuffed into the cranium of a human, and even if there is a sophisticated system of compression and/or multiplexing in use, we can probably assume that a finite amount of axons and dendrites can only carry a finite amount of information.
Thinking of the physical limits on human knowledge leads to a model of knowing that puts the central nervous system at the center of our our experience of the universe. There may be an absolute reality that exists outside of us, but since all of us are human (I assume) we must admit that our knowledge of that absolute reality will forever be shaped by that central nervous system.
The limits of the central nervous system cause humans to look for models of reality that fit well with those limits. For instance, there is The Magical Number Seven, Plus or Minus Two:
http://en.wikipedia.org/wiki/The_Magical_Number_Seven,_Plus_...
Because of the (very very finite) limits on working memory, the human mind leans heavily on heuristics that allow simple models to stand in for the complexity of reality. No human being could easily work with a physics model that had 1 million variables: therefore we tend to look for models that have a handful of variables, yet still give us reasonably good estimates.
Just yesterday there was posted to Hacker News an article about a slinky, and that article contains a good demonstration of the kind of simplifying assumptions we often make to reason about problems that are too complex to approach directly:
"I decided to idealize the problem like this: the slinky is an ideal spring with mass distributed uniformly throughout. It is also a spring that can pass through itself. These assumptions make analyzing the problem easier."
And of course, we often present students with problems where the difficulty of achieving perfect measurements of variables is simply assumed away. For instance in this exercise:
http://www.uta.edu/ce/nsf/ret/docs/lesson-plans/reynolds08-5...
The important things to measure are: force, deflection, torque, moment of inertia, stress and strain. Often, in real life situations, engineers have no way to perfectly measure these things. Large engineering projects proceed with various methods for getting reasonable estimates: known upper and lower bounds, prototypes, isolated testing of particular connections, etc.
The smartest people alive never get to the point where they really know reality, they only know estimates. I do not mean to make this point overly philosophical, I am talking about practicalities here: there may be an absolute reality that we can know directly, but when it comes to doing anything useful, we develop estimates using fairly simple models, models which are well suited to the limits of our minds.
In some sense, the models we build about the world are always simplified models that make it easier for our limited central nervous systems to make estimates about the world.
I just offered a physics examples of simplifying assumptions, but I could also focus on the humanities. Perhaps your life's goal is to figure out the "real" reasons why the French Revolution happened, or why Communism was so popular for so long, or why the Industrial Revolution happened -- the possibilities are as limitless as the combination of all the variables in action over the course of centuries, the psychology and biology of billions of people, and the technology that existed at the time, and the weather and the crops and the economy and the religions and the amount of sunlight reaching the earth and everything else, combining exponentially to some very large sum. You might research the issue for 40 years, but in the end, if you want to draw some useful conclusion, something that others can benefit from, something that might influence policymakers in the future, you will need to come up with a simplified model that focuses on the handful of variables that you regard as the most important.
There is another model of knowing, which takes for granted the absolute reality outside of us. That model is well summarized by this XKCD comic:
http://xkcd.com/435/
In that model, biology is a sub-branch of physics, which is a sub-branch of math. However, if you focus on the limits of the central nervous system as the practical limit of human knowing, then you end up with a model where physics and math (and everything else) is a sub-branch of biology, specifically, a sub-branch of neurology. Even the two models of human knowing that I describe in this post to Hacker News can be thought of as sub-branches of neurology. All human knowing is, in some sense, a clever hack of our neurology, and certainly all heuristics and models are hacks which adapt to the limits of that neurology.
There is nothing wrong with simple models, so long as they allow us to get useful work done. But I think our reliance on simple models tells us why specialization is so important. If a person can study a subject for 40 years and, having read tens of thousands of documents, still distills the data down to a small handful of variables, a model that gracefully adapts to the limits of the human mind, then we must admit that it is the rare human being who goes much beyond simple models. In those cases where simple models don't work, and where there only thing that works is a model with hundreds or thousands of variables, then clearly, achieving an understanding of such models must surely be the work of a lifetime. You need to get to studying when you are 20, and you need to specialize, because when you are 50 you will still be only approaching the ideal, never quite there, only approaching it, still learning more.
I think that our current revolution, the Internet and computers in general, is allowing us to overcome these limits.
The question is how to integrate them better, how to improve the link between humans and computers, and how to improve our own cognition in the most natural, most human way to get the most out of the whole of human knowledge that is now available everywhere.
And for this, we need generalists.
Don't forget writing. Before writing (in some form or another), humans just had to remember things. Writing allowed them to recall things over a much greater period of time, as well as share them.
That was really the first step to working beyond the capacity of our minds.
Writing is like a hard disk. Before that, everything was volatile RAM.
I don't think the question is about how much we can store in our brains, but rather how much we can cram in there within a lifetime. I think we'll always reach death before we fill our brains. Of course, if we ever reach a true singularity, transfer our consciousnesses into robots, and live forever, the one question becomes the other.
And are there natural boundaries and are 3-d space conventional area vs volume concepts relevant if whole new areas open up?
For example there was a relevant fear at one point that if we continued to discover another new chemical element every two decades or so, in a couple centuries no one could possibly know all the elements, perhaps not even name them all. But that growth had a natural border in the triple digit range; no problemo. Of course chemists found entirely new things to spend time on, rather than just discovering new elements.
There may very well be nothing more that can be learned WRT classical mechanics. Plenty of fun quantum and relativistic stuff to learn, but thats not on the "here be dragons" map which is now completely explored for classical mechanics, or as near to completely explored not to matter. Someday those fields will be fully explored and boring and closed to further exploration.
Another interesting topic is eventually stuff is forgotten when its irrelevant. What happens in a million years when most academic activity is reinventing useless stuff thats already been invented, found useless, and forgotten, so how do you stop the backlash from eliminating actually useful knowledge?
Well, a million years is about 5 times the lifetime of our species. Whoever will be reinventing our knowledge will probably be so different from us that this discussion won't make any sense anymore.
A few thousand years ago, somebody probably asked "How will people of the future be able to tell histories once we paint over all the rocks?"
"Whoever will be reinventing our knowledge will probably be so different from us that this discussion won't make any sense anymore."
You're going to make contemporary archeologists cry, saying things like that. And the future ones are likely to be even better at it.
That might make an interesting very long term startup idea, in a million years it may make a heck of a lot more sense to hire archeologists to "discover" new areas for business profit than to hire primary creators to try and re-invent the hard way, the same way the first 345 times "it" was invented. And I'm only slightly tongue in cheek that this is already how contemporary IT works.
In answer to the last question, how about Durkheim's theory of mechanical and organic solidarity? http://en.wikipedia.org/wiki/Mechanical_and_organic_solidari...
Your other comments remind me of what Lagrange said in 1781 about mathematics: "...I am not sure that I shall still do geometry ten years from now. I also think that the mine is already almost too deep, and must sooner or later be abandoned. Today, Physics and Chemistry offer more brilliant discoveries and which are easier to exploit"
http://www.westfield.ma.edu/math/faculty/fleron/quotes/viewq...
Computers to the rescue.