Don’t you mean, “superlinear” ? But you’re right, I should’ve read the full linked article before commenting. Now that I’d read it, though, I am somewhat less than impressed. Here’s one reason for that:
In fact, West’s paper in Science ignited a flurry of rebuttals, in which researchers pointed out all the species that violated the math. West can barely hide his impatience with what he regards as quibbles. “There are always going to be people who say, ‘What about the crayfish?’ ” he says. “Well, what about it? Every fundamental law has exceptions. But you still need the law or else all you have is observations that don’t make sense. And that’s not science. That’s just taking notes.”
Um. If your “fundamental law” has all these exceptions, that’s a good hint that maybe it isn’t as fundamental as you thought. The law of gravity doesn’t have exceptions. And no, it’s not always better to “have the law”. Sometimes it is, for practical reasons, and sometimes it’s better to devise a better law that doesn’t give you so many false positives.
The article goes on to describe the superlinear growth of efficiency in cities, and notes (correctly, IMO) that it cannot be sustained forever:
Because our lifestyle has become so expensive to maintain, every new resource now becomes exhausted at a faster rate. This means that the cycle of innovations has to constantly accelerate, with each breakthrough providing a shorter reprieve...
But I think one point that the article is missing is that cities don’t exist in a vacuum. As a city grows, it requires more food (which can’t be grown efficiently inside the city), more highways (connecting it with its neighbours), etc. If we ignore all of that, we get superlinear scaling; but my guess is that if we include it, we would get sublinear scaling as usual—in terms of overall economic output per single human.
Um. If your “fundamental law” has all these exceptions, that’s a good hint that maybe it isn’t as fundamental as you thought. The law of gravity doesn’t have exceptions. And no, it’s not always better to “have the law”. Sometimes it is, for practical reasons, and sometimes it’s better to devise a better law that doesn’t give you so many false positives.
You’re missing the point too. Even gravity has exceptions—yes, really, this is a standard topic in philosophy of science because the Laws Of Gravity are so clear, yet in practice they are riddled with exceptions and errors. We have errors so large that Newtonians were forced to postulate entire planets to explain them (not all of which turned out as well as Uranus, Neptune, and Pluto), we have errors which took centuries to be winkled out, and of course errors like Mercury which ultimately could be explained only by an entirely new theory.
And we’re talking about real-world statistics: has there ever been a sociology, economics, or biological allometry paper where every single data point was predicted perfectly without any error whatsoever? (If you think this, then perhaps you should consult Tukey and Cohen on how ‘the null hypothesis is always false’.)
If we ignore all of that, we get superlinear scaling; but my guess is that if we include it, we would get sublinear scaling as usual—in terms of overall economic output per single human.
Absolutely; if you measure in certain ways, diminishing returns has clearly set in for humanity. And yet, compared to hunter-gatherers, we might as well be a Singularity.
What does this tell you about the relevance of diminishing returns to Singularity discussions? (Chalmers’s Singularity paper deals with this very question, IIRC, if you are interested in a pre-existing discussion.)
Even gravity has exceptions—yes, really, this is a standard topic in philosophy of science because the Laws Of Gravity are so clear, yet in practice they are riddled with exceptions and errors
In addition to what the others said on this thread, I’d like to say that my main problem was with the author’s attitude, not the accuracy of his proposed law—though the fact that it apparently has glaring holes in it doesn’t really help. When you discover that your law has huge exceptions (such as f.ex. “all crustaceans” or “Mercury”), the thing to do is to postulate hidden planets, or discover relativity, or introduce a term representing dark energy, or something. The thing not to do is to say, “oh well, every law has exceptions, this is good enough for me, case closed ! Let’s pretend that crustaceans don’t exist, we’re done”.
And we’re talking about real-world statistics: has there ever been a sociology, economics, or biological allometry paper where every single data point was predicted perfectly without any error whatsoever?
I’m not sure what you’re referring to; of course, no one expects any line to have a correlation of 1.0 at all times. That’d be silly. However, it is almost equally as silly to take a few data points, and extrapolate them far into the future without any concern for what you’re doing. Ultimately, you can draw a straight line through any two points, but that doesn’t mean that a child will be over 5m tall at age 20 just because he grew 25cm in a year.
Absolutely; if you measure in certain ways, diminishing returns has clearly set in for humanity. And yet, compared to hunter-gatherers, we might as well be a Singularity.
How so ? Perhaps more importantly, if “diminishing returns has clearly set in for humanity” as you say, then what does that tell you for our prospects of bringing about the actual Singularity ?
In addition to what the others said on this thread, I’d like to say that my main problem was with the author’s attitude, not the accuracy of his proposed law—though the fact that it apparently has glaring holes in it doesn’t really help. When you discover that your law has huge exceptions (such as f.ex. “all crustaceans” or “Mercury”), the thing to do is to postulate hidden planets, or discover relativity, or introduce a term representing dark energy, or something. The thing not to do is to say, “oh well, every law has exceptions, this is good enough for me, case closed ! Let’s pretend that crustaceans don’t exist, we’re done”.
Well, that’s useful advice to the Newtonians, alright - ‘hey guys, why did you let the Mercury anomaly linger for decades/centuries? All you had to do was invent relativity! Just ask Bugmaster!’
I wasn’t aware West had retired and was eagerly awaiting his Nobel phone call.
However, it is almost equally as silly to take a few data points, and extrapolate them far into the future without any concern for what you’re doing. Ultimately, you can draw a straight line through any two points, but that doesn’t mean that a child will be over 5m tall at age 20 just because he grew 25cm in a year.
Why do you think the existing dataset is analogous to your silly example?
How so ? Perhaps more importantly, if “diminishing returns has clearly set in for humanity” as you say, then what does that tell you for our prospects of bringing about the actual Singularity ?
Well, that’s useful advice to the Newtonians, alright - ‘hey guys, why did you let the Mercury anomaly linger for decades/centuries? All you had to do was invent relativity! Just ask Bugmaster!’
There’s a difference between acknowledging the problems with your “fundamental law” (once they become apparent, of course) but failing to fix them for “decades/centuries”; vs. boldly ignoring them because “all laws have exceptions, them’s the breaks”. It’s possible that West is not doing the latter, but the article does imply that this is the case.
Why do you think the existing dataset is analogous to your silly example?
Which dataset are you talking about ? If you mean, the growth of cities, then see below.
How so ? Perhaps more importantly, if “diminishing returns has clearly set in for humanity” as you say, then what does that tell you for our prospects of bringing about the actual Singularity ?
Not much.
Why not ? If humanity’s productive output has recently (relatively speaking) reached the point of diminishing returns, then a). we can no longer extrapolate the growth of productivity in cities by assuming past trends would continue indefinitely, and b). this does not bode well for the Singularity, which would entail an exponential growth of productivity, free of any diminishing returns.
It’s possible that West is not doing the latter, but the article does imply that this is the case.
It didn’t sound like that to me. It sounded like some people had absurd standards for scaling phenomena, and he was rightly dismissing them.
If humanity’s productive output has recently (relatively speaking) reached the point of diminishing returns,
There’s nothing recently about it. Diminishing returns is a pretty general phenomenon which happens in most periods; Tainter documents examples in many ancient settings, and we can find data sets suggesting diminishing returns in the West from long ago. For example, IIRC Murray finds that once you adjust for population growth, scientific achievement has been falling since the 1890s or so.
then a). we can no longer extrapolate the growth of productivity in cities by assuming past trends would continue indefinitely, and b). this does not bode well for the Singularity, which would entail an exponential growth of productivity, free of any diminishing returns.
It doesn’t bode much of anything; I referred to you my list of ‘what diminishing returns does not imply’ for a reason: #1-4 are directly relevant. Diminishing returns does not mean no exponential growth; it does not mean no regime changes, massive accomplishments, breakthroughs, or technologies. It just means diminishing returns; it’s just an observation about one unit of input turning into units of output as compared to the previous unit of input and outputs, nothing more and nothing less.
This is obvious if you take Tainter or Murray or any of the results showing any diminishing returns in the past centuries, since those are precisely the centuries in which humanity has done the most extraordinarily well! One could say, with equal justice, that ‘this does not bode well’ for the 20th century; one could say with equal justice in 1950 that diminishing returns bodes poorly for the computer industry because not only are chip fab prices keeping on increasing (‘Moore’s second law’), computing power is visibly suffering diminishing returns as it is applied to more and more worthless problems—where once it was used on problems of vital national value (crucial to the survival of the free world and all that is good) worth billions such as artillery tables and H-bomb simulations, now it was being wasted on grad students and businesses.
Even gravity has exceptions—yes, really, this is a standard topic in philosophy of science because the Laws Of Gravity are so clear, yet in practice they are riddled with exceptions and errors.
No, you did not. Your examples are all consistent with our best current exceptionless theory of gravity (general relativity) and knowledge of the composition of our solar system (Uranus, Neptune, and Pluto). You merely hinted at the existence of additional examples that perplexed the Newtonians. In fact, since our current understanding of gravity is better than the Newtonians’, hinting at the existence of examples that perplexed the Newtonians fails to even suggest a flaw in our best current theory, not to mention suggesting the existence of “exceptions to gravity”. Please give at least one real example.
Nobody brought up relativity as the issue; the fact remains that every theory is incomplete and a work in progress, and a few errors is not disproof especially for a statistical generalization. You would not apply this ultra-high standard of ‘the theory must explain every observation ever in the absence of any further data or modifications’ to anything else discussed on LW, and I do not understand why either you or army1987 think you are adding anything to this discussion about cities exhibiting better scaling than corporations.
You said that gravity has exceptions. I’m not quite sure what that’s supposed to mean, but the only interpretation I could think of for that statement is that our current best theory of gravity (namely, general relativity) fails to predict how gravity behaves in some cases. I did not mean to suggest that any theory must explain every observation correctly to be useful, nor did I mean to imply anything about how well cities and corporations scale. I was merely pointing out that you falsely asserted that you had given examples of exceptions to gravity, when you had in fact you had only given examples of exceptions to Newtonian gravity as it would operate in a solar system similar but not identical to ours.
I have never heard of any observation showing that gravitation as described by general relativity (and, so long as you aren’t very close to something very massive and aren’t travelling at a sizeable fraction of the speed of light, excellently approximated by Newton’s law) might have “exceptions” on Solar System-scale, except possibly the Pioneer anomaly (for which there is a very plausible candidate explanation) and similar. When I read “errors” I hoped you meant measurement uncertainties, but I can’t make sense of the rest of the paragraph assuming you did.
There are no examples of failures of general relativity in that entire article. So far, of the two of you, only army1987 has given an example of an even slightly perplexing observation.
Yes, Tainter is one of a number of sources which are why I think humanity has seen diminishing returns. I’ve been casually dumping some info in http://www.gwern.net/the-long-stagnation although if we were discussing just books, I think Murray’s Human Accomplishment covers convincingly a much more important kind of diminishing returns compared to Tainter’s focus on resources and basic economic metrics.
(For those interested in the topic, I suggest looking at my link just for the intro bit about 5 propositions that the fact of diminishing returns does not prove; I believe more than one commenter on this page is committing at least one of those 5.)
Don’t you mean, “superlinear” ? But you’re right, I should’ve read the full linked article before commenting. Now that I’d read it, though, I am somewhat less than impressed. Here’s one reason for that:
Um. If your “fundamental law” has all these exceptions, that’s a good hint that maybe it isn’t as fundamental as you thought. The law of gravity doesn’t have exceptions. And no, it’s not always better to “have the law”. Sometimes it is, for practical reasons, and sometimes it’s better to devise a better law that doesn’t give you so many false positives.
The article goes on to describe the superlinear growth of efficiency in cities, and notes (correctly, IMO) that it cannot be sustained forever:
But I think one point that the article is missing is that cities don’t exist in a vacuum. As a city grows, it requires more food (which can’t be grown efficiently inside the city), more highways (connecting it with its neighbours), etc. If we ignore all of that, we get superlinear scaling; but my guess is that if we include it, we would get sublinear scaling as usual—in terms of overall economic output per single human.
You’re missing the point too. Even gravity has exceptions—yes, really, this is a standard topic in philosophy of science because the Laws Of Gravity are so clear, yet in practice they are riddled with exceptions and errors. We have errors so large that Newtonians were forced to postulate entire planets to explain them (not all of which turned out as well as Uranus, Neptune, and Pluto), we have errors which took centuries to be winkled out, and of course errors like Mercury which ultimately could be explained only by an entirely new theory.
And we’re talking about real-world statistics: has there ever been a sociology, economics, or biological allometry paper where every single data point was predicted perfectly without any error whatsoever? (If you think this, then perhaps you should consult Tukey and Cohen on how ‘the null hypothesis is always false’.)
Absolutely; if you measure in certain ways, diminishing returns has clearly set in for humanity. And yet, compared to hunter-gatherers, we might as well be a Singularity.
What does this tell you about the relevance of diminishing returns to Singularity discussions? (Chalmers’s Singularity paper deals with this very question, IIRC, if you are interested in a pre-existing discussion.)
In addition to what the others said on this thread, I’d like to say that my main problem was with the author’s attitude, not the accuracy of his proposed law—though the fact that it apparently has glaring holes in it doesn’t really help. When you discover that your law has huge exceptions (such as f.ex. “all crustaceans” or “Mercury”), the thing to do is to postulate hidden planets, or discover relativity, or introduce a term representing dark energy, or something. The thing not to do is to say, “oh well, every law has exceptions, this is good enough for me, case closed ! Let’s pretend that crustaceans don’t exist, we’re done”.
I’m not sure what you’re referring to; of course, no one expects any line to have a correlation of 1.0 at all times. That’d be silly. However, it is almost equally as silly to take a few data points, and extrapolate them far into the future without any concern for what you’re doing. Ultimately, you can draw a straight line through any two points, but that doesn’t mean that a child will be over 5m tall at age 20 just because he grew 25cm in a year.
How so ? Perhaps more importantly, if “diminishing returns has clearly set in for humanity” as you say, then what does that tell you for our prospects of bringing about the actual Singularity ?
Well, that’s useful advice to the Newtonians, alright - ‘hey guys, why did you let the Mercury anomaly linger for decades/centuries? All you had to do was invent relativity! Just ask Bugmaster!’
I wasn’t aware West had retired and was eagerly awaiting his Nobel phone call.
Why do you think the existing dataset is analogous to your silly example?
Not much.
There’s a difference between acknowledging the problems with your “fundamental law” (once they become apparent, of course) but failing to fix them for “decades/centuries”; vs. boldly ignoring them because “all laws have exceptions, them’s the breaks”. It’s possible that West is not doing the latter, but the article does imply that this is the case.
Which dataset are you talking about ? If you mean, the growth of cities, then see below.
Why not ? If humanity’s productive output has recently (relatively speaking) reached the point of diminishing returns, then a). we can no longer extrapolate the growth of productivity in cities by assuming past trends would continue indefinitely, and b). this does not bode well for the Singularity, which would entail an exponential growth of productivity, free of any diminishing returns.
It didn’t sound like that to me. It sounded like some people had absurd standards for scaling phenomena, and he was rightly dismissing them.
There’s nothing recently about it. Diminishing returns is a pretty general phenomenon which happens in most periods; Tainter documents examples in many ancient settings, and we can find data sets suggesting diminishing returns in the West from long ago. For example, IIRC Murray finds that once you adjust for population growth, scientific achievement has been falling since the 1890s or so.
It doesn’t bode much of anything; I referred to you my list of ‘what diminishing returns does not imply’ for a reason: #1-4 are directly relevant. Diminishing returns does not mean no exponential growth; it does not mean no regime changes, massive accomplishments, breakthroughs, or technologies. It just means diminishing returns; it’s just an observation about one unit of input turning into units of output as compared to the previous unit of input and outputs, nothing more and nothing less.
This is obvious if you take Tainter or Murray or any of the results showing any diminishing returns in the past centuries, since those are precisely the centuries in which humanity has done the most extraordinarily well! One could say, with equal justice, that ‘this does not bode well’ for the 20th century; one could say with equal justice in 1950 that diminishing returns bodes poorly for the computer industry because not only are chip fab prices keeping on increasing (‘Moore’s second law’), computing power is visibly suffering diminishing returns as it is applied to more and more worthless problems—where once it was used on problems of vital national value (crucial to the survival of the free world and all that is good) worth billions such as artillery tables and H-bomb simulations, now it was being wasted on grad students and businesses.
What are you talking about?
I gave multiple examples and specified the field interested in how such a naive formulation is completely wrong; please ask better questions.
No, you did not. Your examples are all consistent with our best current exceptionless theory of gravity (general relativity) and knowledge of the composition of our solar system (Uranus, Neptune, and Pluto). You merely hinted at the existence of additional examples that perplexed the Newtonians. In fact, since our current understanding of gravity is better than the Newtonians’, hinting at the existence of examples that perplexed the Newtonians fails to even suggest a flaw in our best current theory, not to mention suggesting the existence of “exceptions to gravity”. Please give at least one real example.
Nobody brought up relativity as the issue; the fact remains that every theory is incomplete and a work in progress, and a few errors is not disproof especially for a statistical generalization. You would not apply this ultra-high standard of ‘the theory must explain every observation ever in the absence of any further data or modifications’ to anything else discussed on LW, and I do not understand why either you or army1987 think you are adding anything to this discussion about cities exhibiting better scaling than corporations.
You said that gravity has exceptions. I’m not quite sure what that’s supposed to mean, but the only interpretation I could think of for that statement is that our current best theory of gravity (namely, general relativity) fails to predict how gravity behaves in some cases. I did not mean to suggest that any theory must explain every observation correctly to be useful, nor did I mean to imply anything about how well cities and corporations scale. I was merely pointing out that you falsely asserted that you had given examples of exceptions to gravity, when you had in fact you had only given examples of exceptions to Newtonian gravity as it would operate in a solar system similar but not identical to ours.
I saw what sounded to me like an extraordinary claim (though it turns out I misunderstood you) so I went WTF.
I have never heard of any observation showing that gravitation as described by general relativity (and, so long as you aren’t very close to something very massive and aren’t travelling at a sizeable fraction of the speed of light, excellently approximated by Newton’s law) might have “exceptions” on Solar System-scale, except possibly the Pioneer anomaly (for which there is a very plausible candidate explanation) and similar. When I read “errors” I hoped you meant measurement uncertainties, but I can’t make sense of the rest of the paragraph assuming you did.
http://en.wikipedia.org/wiki/Philosophy_of_science#Duhem-Quine_thesis may help you a little bit. You should probably read the entire article, since you seem to think there were no errors or exceptions, and that some exceptions could disprove a power law.
I think I know what you mean, but if I’m right, “gravity has exceptions” is, let’s say, a very bizarre way of putting it.
EDIT: yeah, you meant what i thought you meant.
There are no examples of failures of general relativity in that entire article. So far, of the two of you, only army1987 has given an example of an even slightly perplexing observation.
Why should I give one? I never brought up relativity, army1987 did.
You brought up the Laws Of Gravity (capitals yours), which among insiders are known as the Einstein field equations of general relativity.
This seems serendipitous:
http://lesswrong.com/r/discussion/lw/g62/link_the_collapse_of_complex_societies/
Yes, Tainter is one of a number of sources which are why I think humanity has seen diminishing returns. I’ve been casually dumping some info in http://www.gwern.net/the-long-stagnation although if we were discussing just books, I think Murray’s Human Accomplishment covers convincingly a much more important kind of diminishing returns compared to Tainter’s focus on resources and basic economic metrics.
(For those interested in the topic, I suggest looking at my link just for the intro bit about 5 propositions that the fact of diminishing returns does not prove; I believe more than one commenter on this page is committing at least one of those 5.)