It seems to me that “emergence” has a useful meaning once we recognize the Mind Projection Fallacy:
We say that a system X has emergent behavior if we have heuristics for both a low-level description and a high-level description, but we don’t know how to connect one to the other. (Like “confusing”, it exists in the map but not the territory.)
This matches the usage: the ideal gas laws aren’t “emergent” since we know how to derive them (at a physics level of rigor) from lower-level models; however, intelligence is still “emergent” for us since we’re too dumb to find the lower-level patterns in the brain which give rise to patterns like thoughts and awareness, which we have high-level heuristics for.
Thoughts? (If someone’s said this before, I apologize for not remembering it.)
ISTM that the actual present usage of “emergent” is actually pretty well-defined as a cluster, and it doesn’t include the ideal gas laws. I’m offering a candidate way to cash-out that usage without committing the Mind Projection Fallacy.
The fallacy here is thinking there’s a difference between the way the ideal gas laws emerge from particle physics, and the way intelligence emerges from neurons and neurotransmitters. I’ve only heard “emergent” used in the following way:
A system X has emergent behavior if we have heuristics for both a low-level description and a high-level description, and the high-level description is not easily predictable from the low-level description
For instance, gliders moving across the screen diagonally is emergent in Conway’s Life.
The “easily predictable” part is what makes emergence in the map, not the territory.
Yes. My point was that emergence isn’t about what we know how to derive from lower-level descriptions, it’s about what we can easily see and predict from lower-level descriptions. Like Roko, I want my definition of emergence to include the ideal gas laws (and I haven’t heard the word used to exclude them).
For what it’s worth, Cosma Shalizi’s notebook page on emergence has a very reasonable discussion of emergence, and he actually mentions macro-level properties of gas as a form of “weak” emergence:
The weakest sense [i]s also the most obvious. An emergent property is one which arises from the interaction of “lower-level” entities, none of which show it. No reductionism worth bothering with would be upset by this. The volume of a gas, or its pressure or temperature, even the number of molecules in the gas, are not properties of any individual molecule, though they depend on the properties of those individuals, and are entirely explicable from them; indeed, predictable well in advance.
To define emergence as it is normally used, he adds the criterion that “the new property could not be predicted from a knowledge of the lower-level properties,” which looks to be exactly the definition you’ve chosen here (sans map/territory terminology).
If we taboo “emergence” what do we think is going on with Langton’s Ant?
We have one description of the ant/grid system in Langton’s Ant: namely, the rules which totally govern the behavior of the system. We have another description of the system, however: the recurring “highway” pattern that apparently results from every initial configuration tested. These two descriptions seem to be connected, but we’re not entirely sure how (The only explanation we have is akin to this: Q: Why does every initial configuration eventually result in the highway pattern? A: The rules did it.) That is, we have a gap in our map.
Since the rules, which we understand fairly well, seem on some intuitive sense to be at a “lower level” of description than the pattern we observe, and since the pattern seems to depend on the “low-level” rules in some way we can’t describe, some people call this gap “emergence.”
I recall hearing, although I can’t find a link, that the Langton Ant problem has been solved recently. That is, someone has given a formal proof that every ant results in the highway pattern.
It’s worth checking on the Stanford Encyclopedia of Philosophy when this kind of issue comes up. It looks like this view—emergent=hard to predict from low-level model—is pretty mainstream.
The first paragraph of the article on emergence says that it’s a controversial term with various related uses, generally meaning that some phenomenon arises from lower-level processes but is somehow not reducible to them. At the start of section 2 (“Epistemological Emergence”), the article says that the most popular approach is to “characterize the concept of emergence strictly in terms of limits on human knowledge of complex systems.” It then gives a few different variations on this type of view, like that the higher-level behavior could not be predicted “practically speaking; or for any finite knower; or for even an ideal knower.”
There’s more there, some of which seems sensible and some of which I don’t understand.
It seems problematic that as soon as you work out how to derive high-level behavior from low-level behavior, you have to stop calling it emergent. It seems even more problematic that two people can look at the same phenomenon and disagree on whether it’s “emergent” or not, because Bob knows the relevant derivation of high level behavior from low level behavior, but Alice doesn’t, even if Alice nows that Bob knows.
Perhaps we could refine this a little, and make emergence less subjective, but still avoid mind-projection-fallacy.
We say that a system X has emergent behavior if there exists an exact and simple low-level description and an inexact but easy-to-compute high-level description, and the derivation of the high-level laws from the low-level ones is much more complex than either. [In the technical sense of kolmogorov complexity] (Like “Has chaotic dynamics”, it is a property of a system)
I dunno, I kind of like the idea that as science advances, particular phenomena stop being emergent. I’d be very glad if “emergent” changed from a connotation of semantic stop-sign to a connotation of unsolved problem.
By your definition, is the empirical fact that one tenth of the digits of pi are 1s emergent behavior of pi?
I may not understand the work that “low-level” and “high-level” are doing in this discussion.
On the length of derivations, here are some relevant Godel cliches: System X (for instance, arithmetic) often obeys laws that are underivable. And it often obeys derivable laws of length n whose shortest derivation has length busy-beaver-of-n.
I’ll stick to it. It’s easier to perform experiments than it is to give mathematical proofs. If experiments can give strong evidence for anything (I hope they can!), then this data can give strong evidence that pi is normal: http://www.piworld.de/pi-statistics/pist_sdico.htm
Maybe past ten-to-the-one-trillion digits, the statistics of pi are radically different. Maybe past ten-to-the-one-trillion meters, the laws of physics are radically different.
Maybe past ten-to-the-one-trillion digits, the statistics of pi are radically different. Maybe past ten-to-the-one-trillion meters, the laws of physics are radically different.
I was just thinking about the latter case, actually. If g equalled G (m1 ^ (1 + (10 ^ −30)) (m2 ^ (1 + (10 ^ −30))) / (r ^ 2), would we know about it?
Well, the force of gravity isn’t exactly what you get from Newton’s laws anyways (although most of the easily detectable differences like that in the orbit of Mercury are better thought of as due to relativity’s effect on time than a change in g). I’m not actually sure how gravitational force could be non-additive with respect to mass. One would have the problem of then deciding what constitutes a single object. A macroscopic object isn’t a single object in any sense useful to physics. Would for example this calculate the gravity of Earth as a large collection of particles or as all of them together?
But the basic point, that there could be weird small errors in our understanding of the laws of physics is always an issue. To use a slightly more plausible example, if say the force of gravity on baryons is slightly stronger than that on leptons (slightly different values of G) we’d be unlikely to notice. I don’t think we’d notice even if it were in the 2nd or 3rd decimal of G (partially because G is such a very hard constant to measure.)
I have in mind a system, for instance a computer program, that computes pi digit-by-digit. There are features of such a computer program that you can notice from its output, but not (so far as anyone knows) from its code, like the frequency of 1s.
If you had some physical system that computed digit frequencies of Pi, I’d definitely want to call the fact that the fractions were very close to 1 emergent behavior.
I can’t disagree about what you want but I myself don’t really see the point in using the word emergent for a straightforward property of irrational numbers. I wouldn’t go so far as to say the term is useless but whatever use it could have would need to describe something more complex properties that are caused by simpler rules.
This isn’t a general property of irrational numbers, although with probability 1 any irrational number will have this property. In fact, any random real number will have this property with probability 1 (rational numbers have measure 0 since they form a countable set). This is pretty easy to prove if one is familiar with Lebesque measure.
There are irrational numbers which do not share this property. For example,
.101001000100001000001… is irrational and does not share this property.
This isn’t a general property of irrational numbers, although with probability 1 any irrational number will have this property.
Any irrational number drawn from what distribution? There are plenty of distributions that you could draw irrational numbers from which do not have this property, and which contain the same number of numbers in them. For example, the set of all irrational numbers in which every other digit is zero has the same cardinality as the set of all irrational numbers.
Yes, although generally when asking these sorts of questions one looks at the standard Lebesque measure on [0,1] or [0,1) since that’s easier to normalize. I’ve been told that this result also holds for any bell-curve distribution centered at 0 but I haven’t seen a proof of that and it isn’t at all obvious to me how to construct one.
Well, the quick way is to note that the bell-curve measure is absolutely continuous with respect to Lebesgue measure, as is any other measure given by an integrable distribution function on the real line. (If you want, you can do this by hand as well, comparing the probability of a small bounded open set in the bell curve distribution with its Lebesgue measure, taking limits, and then removing the condition of boundedness.)
The only problem with that seems to be that when people talk about emergent behavior they seem to be more often than not talking about “emergence” as a property of the territory, not a property of the map. So for example, someone says that “AI will require emergent behavior”- that’s a claim about the territory. Your definition of emergence seems like a reasonable and potentially useful one but one would need to be careful that the common connotations don’t cause confusion.
I agree. But given that outsiders use the term all the time, and given that they can point to a reasonably large cluster of things (which are adequately contained in the definition I offered), it might be more helpful to say that emergence is a statement of a known unknown (in particular, a missing reduction between levels) than to refuse to use the term entirely, which can appear to be ignoring phenomena.
It seems to me that “emergence” has a useful meaning once we recognize the Mind Projection Fallacy:
We say that a system X has emergent behavior if we have heuristics for both a low-level description and a high-level description, but we don’t know how to connect one to the other. (Like “confusing”, it exists in the map but not the territory.)
This matches the usage: the ideal gas laws aren’t “emergent” since we know how to derive them (at a physics level of rigor) from lower-level models; however, intelligence is still “emergent” for us since we’re too dumb to find the lower-level patterns in the brain which give rise to patterns like thoughts and awareness, which we have high-level heuristics for.
Thoughts? (If someone’s said this before, I apologize for not remembering it.)
No, I want my definition of “emergent” to say that the ideal gas laws are emergent properties of molecules.
Why not just say
We say that a system X has emergent behavior if we have heuristics for both a low-level description and a high-level description
The high-level structure shouldn’t be the same as the low level structure, because I don’t want to say a pile of sand emerges from grains of sand.
ISTM that the actual present usage of “emergent” is actually pretty well-defined as a cluster, and it doesn’t include the ideal gas laws. I’m offering a candidate way to cash-out that usage without committing the Mind Projection Fallacy.
The fallacy here is thinking there’s a difference between the way the ideal gas laws emerge from particle physics, and the way intelligence emerges from neurons and neurotransmitters. I’ve only heard “emergent” used in the following way:
A system X has emergent behavior if we have heuristics for both a low-level description and a high-level description, and the high-level description is not easily predictable from the low-level description
For instance, gliders moving across the screen diagonally is emergent in Conway’s Life.
The “easily predictable” part is what makes emergence in the map, not the territory.
Er, did you read the grandparent comment?
Yes. My point was that emergence isn’t about what we know how to derive from lower-level descriptions, it’s about what we can easily see and predict from lower-level descriptions. Like Roko, I want my definition of emergence to include the ideal gas laws (and I haven’t heard the word used to exclude them).
Also see this comment.
For what it’s worth, Cosma Shalizi’s notebook page on emergence has a very reasonable discussion of emergence, and he actually mentions macro-level properties of gas as a form of “weak” emergence:
To define emergence as it is normally used, he adds the criterion that “the new property could not be predicted from a knowledge of the lower-level properties,” which looks to be exactly the definition you’ve chosen here (sans map/territory terminology).
Let’s talk examples. One of my favorite examples to think about is Langton’s Ant.
If we taboo “emergence” what do we think is going on with Langton’s Ant?
We have one description of the ant/grid system in Langton’s Ant: namely, the rules which totally govern the behavior of the system. We have another description of the system, however: the recurring “highway” pattern that apparently results from every initial configuration tested. These two descriptions seem to be connected, but we’re not entirely sure how (The only explanation we have is akin to this: Q: Why does every initial configuration eventually result in the highway pattern? A: The rules did it.) That is, we have a gap in our map.
Since the rules, which we understand fairly well, seem on some intuitive sense to be at a “lower level” of description than the pattern we observe, and since the pattern seems to depend on the “low-level” rules in some way we can’t describe, some people call this gap “emergence.”
I recall hearing, although I can’t find a link, that the Langton Ant problem has been solved recently. That is, someone has given a formal proof that every ant results in the highway pattern.
It’s worth checking on the Stanford Encyclopedia of Philosophy when this kind of issue comes up. It looks like this view—emergent=hard to predict from low-level model—is pretty mainstream.
The first paragraph of the article on emergence says that it’s a controversial term with various related uses, generally meaning that some phenomenon arises from lower-level processes but is somehow not reducible to them. At the start of section 2 (“Epistemological Emergence”), the article says that the most popular approach is to “characterize the concept of emergence strictly in terms of limits on human knowledge of complex systems.” It then gives a few different variations on this type of view, like that the higher-level behavior could not be predicted “practically speaking; or for any finite knower; or for even an ideal knower.”
There’s more there, some of which seems sensible and some of which I don’t understand.
Many thanks!
It seems problematic that as soon as you work out how to derive high-level behavior from low-level behavior, you have to stop calling it emergent. It seems even more problematic that two people can look at the same phenomenon and disagree on whether it’s “emergent” or not, because Bob knows the relevant derivation of high level behavior from low level behavior, but Alice doesn’t, even if Alice nows that Bob knows.
Perhaps we could refine this a little, and make emergence less subjective, but still avoid mind-projection-fallacy.
We say that a system X has emergent behavior if there exists an exact and simple low-level description and an inexact but easy-to-compute high-level description, and the derivation of the high-level laws from the low-level ones is much more complex than either. [In the technical sense of kolmogorov complexity] (Like “Has chaotic dynamics”, it is a property of a system)
I dunno, I kind of like the idea that as science advances, particular phenomena stop being emergent. I’d be very glad if “emergent” changed from a connotation of semantic stop-sign to a connotation of unsolved problem.
By your definition, is the empirical fact that one tenth of the digits of pi are 1s emergent behavior of pi?
I may not understand the work that “low-level” and “high-level” are doing in this discussion.
On the length of derivations, here are some relevant Godel cliches: System X (for instance, arithmetic) often obeys laws that are underivable. And it often obeys derivable laws of length n whose shortest derivation has length busy-beaver-of-n.
(Uber die lange von Beiwessen is the title of a famous short Godel paper. He revisits the topic in a famous letter to von Neumann, available here: http://rjlipton.wordpress.com/the-gdel-letter/)
Just a pedantic note: pi has not been proven normal. Maybe one fifth of the digits are 1s.
I’ll stick to it. It’s easier to perform experiments than it is to give mathematical proofs. If experiments can give strong evidence for anything (I hope they can!), then this data can give strong evidence that pi is normal: http://www.piworld.de/pi-statistics/pist_sdico.htm
Maybe past ten-to-the-one-trillion digits, the statistics of pi are radically different. Maybe past ten-to-the-one-trillion meters, the laws of physics are radically different.
The later case seems more likely to me.
I was just thinking about the latter case, actually. If g equalled G (m1 ^ (1 + (10 ^ −30)) (m2 ^ (1 + (10 ^ −30))) / (r ^ 2), would we know about it?
Well, the force of gravity isn’t exactly what you get from Newton’s laws anyways (although most of the easily detectable differences like that in the orbit of Mercury are better thought of as due to relativity’s effect on time than a change in g). I’m not actually sure how gravitational force could be non-additive with respect to mass. One would have the problem of then deciding what constitutes a single object. A macroscopic object isn’t a single object in any sense useful to physics. Would for example this calculate the gravity of Earth as a large collection of particles or as all of them together?
But the basic point, that there could be weird small errors in our understanding of the laws of physics is always an issue. To use a slightly more plausible example, if say the force of gravity on baryons is slightly stronger than that on leptons (slightly different values of G) we’d be unlikely to notice. I don’t think we’d notice even if it were in the 2nd or 3rd decimal of G (partially because G is such a very hard constant to measure.)
IMO, that would be emergent behaviour of mathematics, rather than of pi.
Pi isn’t a system in itself as far as I can see.
I have in mind a system, for instance a computer program, that computes pi digit-by-digit. There are features of such a computer program that you can notice from its output, but not (so far as anyone knows) from its code, like the frequency of 1s.
If you had some physical system that computed digit frequencies of Pi, I’d definitely want to call the fact that the fractions were very close to 1 emergent behavior.
Does anyone disagree?
I can’t disagree about what you want but I myself don’t really see the point in using the word emergent for a straightforward property of irrational numbers. I wouldn’t go so far as to say the term is useless but whatever use it could have would need to describe something more complex properties that are caused by simpler rules.
This isn’t a general property of irrational numbers, although with probability 1 any irrational number will have this property. In fact, any random real number will have this property with probability 1 (rational numbers have measure 0 since they form a countable set). This is pretty easy to prove if one is familiar with Lebesque measure.
There are irrational numbers which do not share this property. For example, .101001000100001000001… is irrational and does not share this property.
True enough. it would seem that irrational number is not the correct term for the set I refer to.
The property you are looking for is normalness to base 10. See normal number.
ETA: Actually, you want simple normalness to base 10 which is slighly weaker.
Any irrational number drawn from what distribution? There are plenty of distributions that you could draw irrational numbers from which do not have this property, and which contain the same number of numbers in them. For example, the set of all irrational numbers in which every other digit is zero has the same cardinality as the set of all irrational numbers.
I’m presuming he’s talking about measure, using the standard Lebesgue measure on R
Yes, although generally when asking these sorts of questions one looks at the standard Lebesque measure on [0,1] or [0,1) since that’s easier to normalize. I’ve been told that this result also holds for any bell-curve distribution centered at 0 but I haven’t seen a proof of that and it isn’t at all obvious to me how to construct one.
Well, the quick way is to note that the bell-curve measure is absolutely continuous with respect to Lebesgue measure, as is any other measure given by an integrable distribution function on the real line. (If you want, you can do this by hand as well, comparing the probability of a small bounded open set in the bell curve distribution with its Lebesgue measure, taking limits, and then removing the condition of boundedness.)
Excellent, yes that does work. Thanks very much!
The only problem with that seems to be that when people talk about emergent behavior they seem to be more often than not talking about “emergence” as a property of the territory, not a property of the map. So for example, someone says that “AI will require emergent behavior”- that’s a claim about the territory. Your definition of emergence seems like a reasonable and potentially useful one but one would need to be careful that the common connotations don’t cause confusion.
I agree. But given that outsiders use the term all the time, and given that they can point to a reasonably large cluster of things (which are adequately contained in the definition I offered), it might be more helpful to say that emergence is a statement of a known unknown (in particular, a missing reduction between levels) than to refuse to use the term entirely, which can appear to be ignoring phenomena.