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I'm wondering if there is a general consensus on whether there are emergent properties here, and if there is some consensus that there is, what is a general account of which properties are emergent (molecular properties not due to properties of the constitutive atoms, consciousness, etc.).

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I'm wondering if there is a general consensus on whether there are emergent properties here, and if there is some consensus that there is, what is a general account of which properties are emergent (molecular properties not due to properties of the constitutive atoms, consciousness, etc.).

My guess would be 'yes' to the first question, 'no' to the second.

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I'm wondering if there is a general consensus on whether there are emergent properties here, and if there is some consensus that there is, what is a general account of which properties are emergent (molecular properties not due to properties of the constitutive atoms, consciousness, etc.).

I would call "emergent" that property of a system which is not found in, or posessed by, its constituents.

Examples:

- temperature and entropy; they characterize macroscopic systems only, are not found in individual atoms or molecules, but are due to their movement

- DNA-replication; not found in individual molecules, but is due to their various characteristics

Your definition, "property not due to properties of the constitutive atoms", is not precise enough, in my view.

Alex

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Yeah, I didn't want to ask a more precise question because I didn't want to ask about one particular theory of emergence.

It sounds, perhaps, as though the two of you don't believe there are genuinely emergent properties (emergent in the strong sense, that there are properties which are not reducible to fundamental properties operating in concert by elementary operations). Perhaps you believe only that emergence is a useful way of thinking and organizing information.

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Yeah, I didn't want to ask a more precise question because I didn't want to ask about one particular theory of emergence.

It sounds, perhaps, as though the two of you don't believe there are genuinely emergent properties (emergent in the strong sense, that there are properties which are not reducible to fundamental properties operating in concert by elementary operations). Perhaps you believe only that emergence is a useful way of thinking and organizing information.

I do believe that there exist properties of systems which are not found in, or possessed by, its constituents.

I also believe that they are due to, or are dependent on, the properties of the underlying components, but are not "reducible" to those properties (depending, however, on what does "reducible" means exactly.)

What does "operating in concert by elementary operations" means, do not know.

I do not think that scientific concepts (or science itself) are but a "useful way of thinking and organizing information".

To sum up: I do believe the concept of emergent properties, such as I defined it, is a valid concept. If you believe it is not, you should tackle my examples.

There are, certainly, different concepts of emergent properties, and some of them might be invalid. For example, if emergent property means the one that is in no way dependent on any of the characteristics of the microscopic components (one characteristic being their existence), then such a concept is invalid.

Alex

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Though I'm pretty sure you're not saying this, just to be perfectly clear, you don't mean to say that scientific concepts are entirely mental--that they describe nothing in the objective world, do you?

When I define emergent properties as those not reducible to fundamental properties operating in concert by elementary operations, take the example of the ridges that you see in the sand in a desert. This doesn't seem to be a candidate for emergence of this sort because every event that led to the formation of the ridges seems to be perfectly described in terms of the position, velocity, and mass of each grain of sand, molecule of air, etc. Given sufficient information and using Newtonian mechanics, in principle you can explain all of the events from the history to the present. (Forgetting complications of quantum mechanics.) These Newtonian mechanics are the elementary operations (though they need not be the only elementary operations you might consider in the context of some other example).

Something which is often suspected of being emergent are the properties of molecules, or human consciousness. It is claimed that the properties of molecules cannot, even in principle, be explained only in terms of the properties of the atoms which make it up--you need new, categorically distinct properties to explain the behavior of molecules.

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When I define emergent properties as those not reducible to fundamental properties operating in concert by elementary operations, take the example of the ridges that you see in the sand in a desert. This doesn't seem to be a candidate for emergence of this sort because every event that led to the formation of the ridges seems to be perfectly described in terms of the position, velocity, and mass of each grain of sand, molecule of air, etc.
Do you mean events which, at present, we know how to fully describe in terms of "fundamental properties" and "elementary operations". Or are you looking for events which are claimed to be beyond the possibility of ever explaining in terms of "fundamental properties" and "elementary operations".

What is a "fundamental properties", and what are the "elementary operations"? Here's an illustration of why I care. In grammar, there are certain "fundamental properties" such as "type" (might be "noun", "verb" etc), "direction" (left, right), "distance" (local, unbounded), and "elementary operations" like move, delete, copy. These allow you to map between "He has eaten" and "Has he eaten?". The properties of questions are therefore, by that charaacterization, not "emergent". But if you reject my fundamental properties and elementary operations (and don't substitute alternative fundamental properties and elementary operations of your own), then by your definition, properties of questions would be "emergent".

I assume that you would accept something like my set of fundamentals, since you allow "reduction to Newtonian mechanics" to thwart emergentism (and you know that Newtonian mechanics is, in fact, wrong). That suggest to me that you consider being "emergent" to be a basically epistemological construct -- that is certainly how I understand it -- meaning basically "outside the realm of explanation by the theory".

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I'm wondering if there is a general consensus on whether there are emergent properties here, and if there is some consensus that there is, what is a general account of which properties are emergent (molecular properties not due to properties of the constitutive atoms, consciousness, etc.).

An emergent property is a property of a whole which can never be a property of any part. Example: being alive can never be an attribute of any atom or particle. Mass is not an emergent property because the mass of a whole is simply the sum of the masses of the parts.

I think the perspective that this is merely an epistemological issue is in error. Calling something emergent because we don't have an explanation for it accomplishes nothing, and can be refuted by the discovery of an explanation. I further don't think the "not reducible" standard is good enough either because everything is reducible in principle.

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Do you mean events which, at present, we know how to fully describe in terms of "fundamental properties" and "elementary operations". Or are you looking for events which are claimed to be beyond the possibility of ever explaining in terms of "fundamental properties" and "elementary operations".

What is a "fundamental properties", and what are the "elementary operations"? Here's an illustration of why I care. In grammar, there are certain "fundamental properties" such as "type" (might be "noun", "verb" etc), "direction" (left, right), "distance" (local, unbounded), and "elementary operations" like move, delete, copy. These allow you to map between "He has eaten" and "Has he eaten?". The properties of questions are therefore, by that charaacterization, not "emergent". But if you reject my fundamental properties and elementary operations (and don't substitute alternative fundamental properties and elementary operations of your own), then by your definition, properties of questions would be "emergent".

I assume that you would accept something like my set of fundamentals, since you allow "reduction to Newtonian mechanics" to thwart emergentism (and you know that Newtonian mechanics is, in fact, wrong). That suggest to me that you consider being "emergent" to be a basically epistemological construct -- that is certainly how I understand it -- meaning basically "outside the realm of explanation by the theory".

So the question of emergentism is neutral on which properties and operations you take to be fundamental or elementary. In your example, if what you are describing are the generation of words out of, say, a computer, then your operations would be fundamental. For any word, phrase, or sentence, you could use the alphabet along with a finite list of other characters (like a space, comma, etc.) to generate everything. However, they would not be fundamental if you are talking about the operations that are used to form new words, concepts, sentences.

The point is merely that, in a context, once you have settled on what are your most basic operations (or at least some relatively basic operations), you call "emergent" those properties which are caused by a certain combination of these more basic properties, but which cannot be described purely in terms of them.

An emergent property is a property of a whole which can never be a property of any part. Example: being alive can never be an attribute of any atom or particle. Mass is not an emergent property because the mass of a whole is simply the sum of the masses of the parts.

I think the perspective that this is merely an epistemological issue is in error. Calling something emergent because we don't have an explanation for it accomplishes nothing, and can be refuted by the discovery of an explanation. I further don't think the "not reducible" standard is good enough either because everything is reducible in principle.

Let's be careful. Emergent properties are more than just those which cannot be predicated of a smaller part. For example, nobody takes this to be an emergent property: One man cannot lift a very heavy object, and another man cannot lift it, but the two together can lift it, thus the two together have the property of being able to lift the object while neither in isolation does. This kind of behavior can be reduced to (relatively) more fundamental operations working in concert, namely, exerting a certain amount of force respectively.

So the question is not: Can we say that any one atom or molecule, or brain cell is alive? It is: Can life be described purely in terms of molecules and cells, or at some point, do we need a unique and stratified property which emerges from the more fundamental properties? Your very last sentence suggests that you don't believe in emergentism in the strong sense, while complexity theorists seem to believe in it.

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So the question of emergentism is neutral on which properties and operations you take to be fundamental or elementary.
Okay, so this is basically an epistemological concept (as I've thought), and therefore yes, there are huge numbers of emergent phenomena. Basically, anything that is outside the scope of explanation in your theory.
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Wikipedia: "The way complex systems and patterns arise out of a multiplicity of relatively simple interactions".

That fits everything in the universe above the level of quarks. So pointing to a thing and saying "That's not a quark!" doesn't feel like an explanation to me. I avoid "emergent phenomena" as it's not an explanation of anything. I want to note I don't have a problem with the verb "emerge", as in "Y emerges from X", where X is some detailed model with interactions.

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I want to note I don't have a problem with the verb "emerge", as in "Y emerges from X", where X is some detailed model with interactions.
Is this different from saying "is a consequence"? If not, then I have a problem with saying "emerge", because the exact meaning of "emerge" in this context is obscure.
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Okay, so this is basically an epistemological concept (as I've thought), and therefore yes, there are huge numbers of emergent phenomena. Basically, anything that is outside the scope of explanation in your theory.

So I believe that the epistemological understanding of emergence is uncontroversially correct, but that there is no evidence for metaphysically emergent properties, and they may be theoretically undesirable.

Here is the most thorough and focused discussion of emergence I have found outside of a text: http://plato.stanford.edu/entries/properties-emergent/

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Let's be careful. Emergent properties are more than just those which cannot be predicated of a smaller part. For example, nobody takes this to be an emergent property: One man cannot lift a very heavy object, and another man cannot lift it, but the two together can lift it, thus the two together have the property of being able to lift the object while neither in isolation does. This kind of behavior can be reduced to (relatively) more fundamental operations working in concert, namely, exerting a certain amount of force respectively.

So the question is not: Can we say that any one atom or molecule, or brain cell is alive? It is: Can life be described purely in terms of molecules and cells, or at some point, do we need a unique and stratified property which emerges from the more fundamental properties? Your very last sentence suggests that you don't believe in emergentism in the strong sense, while complexity theorists seem to believe in it.

"One man cannot lift a very heavy object" is not an attribute of a single entity. The better analysis of your lifting example is that the net lifting force on an object is the sum of all the individual forces exerted on the object. This is just like the mass example I gave above in that the whole force is distributed among the parts. I agree that the threshold which appears in the lifting example (is the total lifting force greater than the total downward force?) is not a justification to resort to using 'emergence'.

'Incommensurate' is a distinguishing feature to keep in mind in forming this concept's definition. Any candidate emergent property should be incommensurate with all of the properties of its parts. The life of a multicellular organism is not emergent with respect to its cells, but the life of a cell is emergent with respect to its molecules and atoms. Even if a complete explanation can be given for deriving an emergent property from its parts, if the emergent property is still incommensurate it remains emergent.

An example of an emergent property for which a complete explanation is known is temperature. Individual atoms and molecules do not have temperatures, merely velocities. There is even a statistical correlation between temperature and mean speed of the constituent particles. An analysis in terms of energy is not able to wipe out the emergence. A bullet in flight and the same bullet at rest but heated can both contain the same amount of energy, the difference is in the correlation of the directions of the velocities of the individual particles. A correlation between particles can never be an attribute of a single particle, and this incommensurability is the basis for the emergent nature of temperature and the distinction between energy and temperature.

edit: 'mean speed' for 'mean velocity'

Edited by Grames
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Though I'm pretty sure you're not saying this, just to be perfectly clear, you don't mean to say that scientific concepts are entirely mental...

Concepts are entirely mental.

...you don't mean they describe nothing in the objective world, do you?

It depends; the valid ones do describe something in the objective world.

When I define emergent properties as those not reducible to fundamental properties operating in concert by elementary operations,

I repeat: I do not understand what "operating in concert by elementary operations" means. I am not familiar with this terminology.

take the example of the ridges that you see in the sand in a desert. This doesn't seem to be a candidate for emergence of this sort because every event that led to the formation of the ridges seems to be perfectly described in terms of the position, velocity, and mass of each grain of sand, molecule of air, etc. Given sufficient information and using Newtonian mechanics, in principle you can explain all of the events from the history to the present.

This is a misconception, although it is widely held.

But first, a caveat: one does not need to invoke the possibility of computing something (using Newtonian mechanics or anything else) in order to argue for causality. Yes, the ridges are what they are and could not be different, given the nature of the entities involved. However, I will mention for completeness, that there are other types of "events from the history to the present" which could have been different: these are those which are influenced by a consciousness which has free will.

Now back to "explaining all the events by using Newtonian mechanics". A fact about the Newtonian mechanics is that the solution of the corresponding equations are extremely sensitive with respect to the initial positions, velocities, masses, forms and material of the particles. By this I mean the following: if one makes a change on the 100th decimal place of any of these characteristics, of any of the particles, after a time the equations will give a completely different state. Depending on the nature of such a many-body systems, this time may already be milliseconds.

To say nothing of the fact that one cannot know, and process, the initial characteristics of particles with infinite precision (unless being omniscient, and omnipotent), of the fact that the equations are not exact, that the models for the particles are not exact, etc., etc. To consider, under such circumstances, that some specific equations do permit to correctly compute ("explain"), even in principle, the final state of a complex system, is an illusion and a fallacy. The above is not specific to the Newtonian mechanics: any mathematical description of a many-body system will exhibit such instabilities. For this reason the description in terms of different, collective, properties, will always be indispensable in principle.

That's it for now

Alex

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"One man cannot lift a very heavy object" is not an attribute of a single entity.

I don't understand. One man is a single entity, and of him you predicate the ability to life a heavy object.

The life of a multicellular organism is not emergent with respect to its cells, but the life of a cell is emergent with respect to its molecules and atoms. Even if a complete explanation can be given for deriving an emergent property from its parts, if the emergent property is still incommensurate it remains emergent.

I don't understand this either, but that is perhaps because I am very uneducated about biology. Let us try a thought experiment, though: Suppose you completely describe a living organism, except that you make no explicit reference to life. Instead, you just talk about chemicals and their bonds, with such detail that when you finish your description, you have told everything that need be said to specify some particular organism. What then does talk about "life" contribute? Is this just something which is useful for humans to think about, so that they can abstract about general behaviors of living things?

Or is it that, instead of just being something useful for limited human intelligence, you are actually still missing something in your description if you haven't talked about life? For instance, say that you want to predict the future behavior of the thing, and you try to do it completely in terms of its chemical description without reference to "life". Will you be able to talk about all the things that it can or will do, without talking about it in terms of life? Or does life provide a NEW property, with NEW causal powers, which are not contained just in the properties of the chemicals?

An example of an emergent property for which a complete explanation is known is temperature. Individual atoms and molecules do not have temperatures, merely velocities.

Here again, it seems like talk of temperature is just a way for human to conceputalize more basic physical behavior. If you gave a complete description of the position and velocities of the molecules, you would not need to talk about "temperature" except that it's more convenient. But it doesn't actually add information, once you know all of the information about the smaller parts.

To consider, under such circumstances, that some specific equations do permit to correctly compute ("explain"), even in principle, the final state of a complex system, is an illusion and a fallacy. The above is not specific to the Newtonian mechanics: any mathematical description of a many-body system will exhibit such instabilities. For this reason the description in terms of different, collective, properties, will always be indispensable in principle.

I've been getting that kind of response from some scientists, but I don't understand why. We'll omit conversation about free will for the sake of simplicity and not needlessly making a subject more controversial than it already is. Let's just talk about inanimate matter, maybe organisms without a central nervous system.

But let us suppose that you get very, very precise measurements of an initial condition. Then you do a bunch of guessing in order to make it more precise, but it just turns out that you're guesses are completely and perfectly accurate, even into the infinite decimal expansion. It seems like, in this case, you are able to use your theory to predict every event. Even though you didn't use a reliable method of measurement, you still got the numbers that give a perfect description.

Naturally, QM may throw this off, but I'm just talking within the assumptions of Newtonian mechanics.

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I don't understand. One man is a single entity, and of him you predicate the ability to life a heavy object.

That is a relationship between two entities. Emergence is about entities comprised of parts, i.e. internal relations. What super-entity has as its parts a man and the thing he is lifting?

I don't understand this either, but that is perhaps because I am very uneducated about biology. Let us try a thought experiment, though: Suppose you completely describe a living organism, except that you make no explicit reference to life. Instead, you just talk about chemicals and their bonds, with such detail that when you finish your description, you have told everything that need be said to specify some particular organism. What then does talk about "life" contribute? Is this just something which is useful for humans to think about, so that they can abstract about general behaviors of living things?

Or is it that, instead of just being something useful for limited human intelligence, you are actually still missing something in your description if you haven't talked about life? For instance, say that you want to predict the future behavior of the thing, and you try to do it completely in terms of its chemical description without reference to "life". Will you be able to talk about all the things that it can or will do, without talking about it in terms of life? Or does life provide a NEW property, with NEW causal powers, which are not contained just in the properties of the chemicals?

From an imagined impersonal and omniscient perspective the concept life adds nothing. That perspective is not available to us, so 'life' is useful (and not arbitrary). I have to take issue with the perspective behind the question and its phrasing. "Just useful for humans"? Is that an insufficient standard? Is there a higher truth or superior being whose methods toward which we should aspire? :P

Here again, it seems like talk of temperature is just a way for human to conceputalize more basic physical behavior. If you gave a complete description of the position and velocities of the molecules, you would not need to talk about "temperature" except that it's more convenient. But it doesn't actually add information, once you know all of the information about the smaller parts.

Concepts are ways to omit information, not add it. The reduction of information is precisely what makes the concept useful. The incommensurate property of 'correlation' is the existent that makes the concept scientifically valid and not arbitrary.

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I've been getting that kind of response from some scientists, but I don't understand why.

You mean you don't understand the argument? It is contained in the part you omitted:

A fact about the Newtonian mechanics is that the solution of the corresponding equations are extremely sensitive with respect to the initial positions, velocities, masses, forms and material of the particles. By this I mean the following: if one makes a change on the 100th decimal place of any of these characteristics, of any of the particles, after a time the equations will give a completely different state. Depending on the nature of such a many-body systems, this time may already be milliseconds.

To say nothing of the fact that one cannot know, and process, the initial characteristics of particles with infinite precision (unless being omniscient, and omnipotent), of the fact that the equations are not exact, that the models for the particles are not exact, etc., etc.

... let us suppose that you get very, very precise measurements of an initial condition.

OK, let's assume you determined the initial position of the center of mass of a grain of sand with 10 decimals.

Then you do a bunch of guessing in order to make it more precise, but it just turns out that you're guesses are completely and perfectly accurate, even into the infinite decimal expansion. It seems like, in this case, you are able to use your theory to predict every event. Even though you didn't use a reliable method of measurement, you still got the numbers that give a perfect description.

This phrase is a mystery to me. Would you care to explain?

Naturally, QM may throw this off, but I'm just talking within the assumptions of Newtonian mechanics.

1. If Quantum, and not Classical Mechanics (CM) is known to be the correct theory, than thinking within the assumptions of CM is a useless exercise. Besides, neither the QM is the correct theory.

2. But you know what? Computing the movement of the sand particles by QM will not throw you off more than CM; it will just add to the complexity of computation, but will display the same sensitivity with respect to the initial conditions and to the other assumption.

It is a persistent misconception that QM introduces so called incertitudes. Firstly, micro-particles just do not posses sharp positions and velocities. Then: for the grain of sand - a macroscopic entity - the sharp position and velocity are emergent :-) properties. And finally, QM does not describe a grain of sand better (= more precisely) than CM, just that the math is much complexer.

So: I wait for you to explain the phrase of which I said that it was a mystery to me.

Alex

PS: If you are looking for properties of a macroscopic system which are not sensitive to (or dependent of) the initial conditions for its constituents, than they are... the emergent :P ones, like volume, (local) pressure, temperature and such. And this can be proved: it's called the ergodic theorem.

Edited by AlexL
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That is a relationship between two entities. Emergence is about entities comprised of parts, i.e. internal relations. What super-entity has as its parts a man and the thing he is lifting?

Emergence is also about causal power--properties, I believe, are effectively descriptions of causal power. (To say that something is wet will communicate that if you touch it, it will have a particular texture.) So it is also about external relations. For instance, the supposed emergent properties of a molecule, perhaps that of volatility, will cause it to--perhaps--release energy when in contact with another molecule.

But if you want to talk about such super-entities, I still wouldn't mind talking about that entity which is a man and the thing he is lift. I believe in unrestricted composition.

From an imagined impersonal and omniscient perspective the concept life adds nothing.

I think this implies that you disagree with emergence as complexity theorists present it (the strong thesis), and I believe I disagree with it as well. The strong thesis would imply that the property of life, if it is emergent, is something new and unique which springs out of the more fundamental properties, but which cannot be reduced to them--and so any description of a living organism in terms of its cells, no matter how perfect, will never fully describe the organism.

Concepts are ways to omit information, not add it. The reduction of information is precisely what makes the concept useful. The incommensurate property of 'correlation' is the existent that makes the concept scientifically valid and not arbitrary.

I believe I have been working completely within the frame of properties rather than concepts.

This phrase is a mystery to me. Would you care to explain?

So you determine the location of the center of mass of a grain of sand to an incredible degree. But it's not perfect. So you guess as to its exact degree and, unbeknownced to you this guess gives the exact location.

1. If Quantum, and not Classical Mechanics (CM) is known to be the correct theory, than thinking within the assumptions of CM is a useless exercise. Besides, neither the QM is the correct theory.

The point is just to look at an example of something which is not emergent, to help determine what emergence is, so we assume CM for the sake of conversation.

It is a persistent misconception that QM introduces so called incertitudes. Firstly, micro-particles just do not posses sharp positions and velocities.

I don't know what you mean by "introduce". It sounds like you think it's a persistent misconception that QM creates, rather than describes, uncertainty. As if the theory came along and all of the sudden there is uncertainty in the world, like saying the magic words "quantum mechanics" brought it into being. Which I doubt anyone holds. Otherwise it seems like most people (informed enough to discuss the matter) realize that QM's claim is that micro-particles do not posses sharp positions and velocities.

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Is this different from saying "is a consequence"? If not, then I have a problem with saying "emerge", because the exact meaning of "emerge" in this context is obscure.

I'll be more explicit. I don't mind the verb emerge when it's synonymous with "arises from", or "is caused by". e.g. Gravity arises from the curvature of spacetime, according to the very specific mathematical model of General Relativity. The bad emergence is if I said gravity is explained by "arisence", or "arising phenomena." Easy test to prove its worthlessness: replace "emergence" with nothing, and see if the sentence means anything different. Or even more fun, replace it with the old word "magic".

Before: Life is an emergent phenomenon.

After: Life is a magical phenomenon.

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Emergence is also about causal power--properties, I believe, are effectively descriptions of causal power. (To say that something is wet will communicate that if you touch it, it will have a particular texture.) So it is also about external relations. For instance, the supposed emergent properties of a molecule, perhaps that of volatility, will cause it to--perhaps--release energy when in contact with another molecule.

The molecule is the super-entity here, the atoms are the parts. The emergent property here is an attribute of the molecule caused by the internal relations of the parts. Other molecules are not the cause of the attributes of this molecule.

I believe in unrestricted composition.

I believe you had mentioned that bizarre doctrine in another thread. I reject it and don't want to discuss it here.

I believe I have been working completely within the frame of properties rather than concepts.

I believe you are posting on ObjectivismOnline.net so an analysis based on Objectivist concept formation is appropriate.

Temperature is a property and it is a concept. Concepts are epistemological and valid concepts have referents that are existents. Attributes identified as emergent have to be both real and somehow distinct from other attributes. Temperature is both real and distinct from the energy of a single particle. A collection of single particles does not behave differently if a temperature is computed or measured because temperature is derivative, dependent and emergent from multiple particles. Temperature does not add anything to the reality of the single particles, it is a way of regarding those particles in thought.

No frame can disentangle the concept of emergent properties without a sound theory of concepts to accompany the understanding of what happens. If you pretend to take on the perspective of a mentality that deals with reality in its full detail so does not need concepts then you don't need emergence either.

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The molecule is the super-entity here, the atoms are the parts. The emergent property here is an attribute of the molecule caused by the internal relations of the parts. Other molecules are not the cause of the attributes of this molecule.

Nobody said the other molecule caused the property, nor did I say in the original example that the heavy object caused the property.

I believe you are posting on ObjectivismOnline.net so an analysis based on Objectivist concept formation is appropriate.

Only if I'm talking about a concept and not a property. Even on ObjectivismOnline.net, I expect to discuss the subject matter, rather than the nearest convenient subject such that it conforms to Objectivist language and doctrine.

Temperature is a property and it is a concept.

Temperature is a property and not a concept. A thing that is hot is not also conceptually hot. There is, however, a concept of temperature, if you want to split hairs.

In any case, it seems I've got you down for one rejection of emergentism (in the strong sense).

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