August 17, 1994: "Representing Emergence with Rules: The Limits of Addition." Presented at the 7th International Conference on Systems Research, Information and Cybernetics. Sponsored by The International Institute for Advanced Studies in Systems Research and Cybernetics, and the Society for Applied Systems Research. Paper published in Lasker, G. E. and Farre, G. L. (editors), Advances in Synergetics, Volume I: Systems Research on Emergence. (1994)
Cover Page Uploaded June 22, 2011 Representing Emergence with Rules Author: Jeffrey G. Long ([email protected]) Date: August 17, 1994 Forum: Talk presented at the 7th International Conference on Systems Research, Information and Cybernetics. Sponsored by The International Institute for Advanced Studies in Systems Research and Cybernetics, and the Society for Applied Systems Research. Paper published in conference proceedings, available at http://iias.info/pdf_general/Booklisting.pdf Contents Pages 1‐6: Abstract and Preprint of paper Pages 7‐24: Slides but no text of oral presentation License This work is licensed under the Creative Commons Attribution‐NonCommercial 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by‐nc/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA.
Forum: Talk presented at the 7th International Conference on Systems Research, Information and Cybernetics. Sponsored by The International Institute for Advanced Studies in Systems Research and Cybernetics, and the Society for Applied Systems Research. Paper published in conference proceedings, available at http://iias.info/pdf_general/Booklisting.pdf
Contents
Pages 1‐6: Abstract and Preprint of paper
Pages 7‐24: Slides but no text of oral presentation
License
This work is licensed under the Creative Commons Attribution‐NonCommercial
3.0 Unported License. To view a copy of this license, visit
http://creativecommons.org/licenses/by‐nc/3.0/ or send a letter to Creative
Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA.
Abstract Emergence may be defined as the point at which an entity is subject to a new and different class of rules. Given that we can describe entities at one level (e.g. the properties of hydrogen and oxygen) in terms of their probable rules of behavior, and can describe entities at a higher level (e.g. the properties of water) in the same manner, the essential question of emergence becomes "How does an entity become subject to a completely new and different set of rules?" This paper describes the notion of emergence operationally, by means of a very simple "emergence rule" that declares the existence of new entities whenever existing entities achieve certain defined statuses. Conversely, any time an entity becomes subject to a new and different class of rules, it operationally becomes a new entity. Entities change statuses only as a by‐product of processes, which processes can perform only "addition" or "subtraction" in the broadest senses of the words. Under this approach, there exist two broad classes of phenomena: those that follow the classical rules of arithmetic (called resultants), and those that don't (called emergents). Both of these may be described by means of qualitative, conditional rules. The paper illustrates these concepts with examples of emergence from intentional systems (the U.S. Constitution) and natural systems (basic chemistry). Keywords emergence; processes; limits of mathematics; rules; notation; law Introduction The paradoxes of complexity, and in particular the phenomena of emergence, have forced me to reconsider how we represent the basic and ubiquitous transactions of addition and subtraction. My conclusion to date is that we must create another "grammar" that distinguishes resultant from emergent1 transactions. The class of transactions that I will call resultant transactions causes no real difficulties for modern quantitative notation. These transactions arise when two or more entities can be "added together" in any order, and their interaction is zero or negligible. In these transactions, one plus one equals two, now and forever; in other words, they are additive simpliciter. In pure arithmetic, this denotes the union of two sets; in the real world, it may refer to two entities being mixed, or placed nearby each other, or in some other sense "added". The notational systems that have been built around numbers presume the following:
entities in toto can be added or subtracted simpliciter in all interesting cases
properties of entities can be added or subtracted simpliciter in all interesting cases
1 -- See G. H. Lewes, who proposed the terminology.
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such transactions are commutative (i.e., that adding A to B is identical in effect to adding B to A)
they are associative (i.e., that adding A to B and then to C is identical to adding B to C and then to A)
they are monotonic (i.e. if A < B, then A + X < B + X)
they are reversible (i.e. that if A has been added to B, then A can be subtracted from B to derive separate entities again). These transactional presumptions hold true of all vector spaces, i.e. all sets "of objects or elements that can be added together and multiplied by numbers (the result being an element of the set), in such a way that the usual rules of calculation hold" (Gellert et al, page 362). They work quite well for the natural numbers, for many other kinds of mathematical entity (e.g. angles), and for many entities in the real world (e.g. unbalanced forces); but many transactions in the real world do not fit these criteria. Such transactions ‐‐ for which I will hijack the phrase emergent transactions ‐‐ occur when two or more entities are "added together" or "subtracted", and their interaction is significant. There are several signs that indicate when an emergent transaction has occurred:
the resulting number of entities cannot be inferred from the number of the components (the quantity of the sum cannot be computed from the quantities of the summands)(i.e., a unit increase in summands produces a non‐unit increase in the sum2)
the resulting properties cannot be inferred from the properties of the components (the properties of the sum cannot be computed from the properties of the summands)
the order of addition (or subtraction) is significant and is not reversible. The transactions that have created matter, life, society, consciousness ‐‐ and perhaps even notations like number3 ‐‐ are emergent transactions. The classic example is the combination of two flammable gasses ‐‐ hydrogen and oxygen ‐‐ to form a non‐flammable liquid: water. An example of non‐commutativity is the addition of water to acid versus the addition of acid to water.
One of the seminal thinkers about emergent evolutionism, C. Lloyd Morgan, suggested that emergent transactions produce qualitative changes; but I observe that they can also create quantitative changes. Conversely, resultant transactions also produce qualitative changes, such as the addition of blue and yellow watercolors to make green watercolor. Figure 1 illustrates the key distinctions between a resultant grammar of interactions and an emergent grammar:
Qualitative A + B = "AB" "ABC"‐B= "AC" A + B = C C ‐ B = A, B
Figure 1: Two Grammars of Interaction
2 -- Thus non-linear transactions are a subset of emergent transactions. 3 -- In January 1994 I gave a talk exploring the idea that notations are real in the Platonic sense of being pre-existing rather than emergent. I confess I am presently confused about my beliefs in this area.
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Thus the key distinction between resultant and emergent transactions is not whether they are quantitative or qualitative, but whether they are predictable. Emergent transactions do seem to violate the traditional and common‐sense beliefs that every effect has a cause, and that full knowledge of causes permits one to deduce or predict all of their effects. Bertrand Russell, in support of these beliefs, asserted that "Emergent properties represent merely scientific incompleteness, which would not exist in the ideal physics" (Russell, 1929). Whatever the causality issues may be, if we can simply describe our empirical experience accurately we will have a far greater understanding of the world we live in. Representation Issues If we look at the universe as changing at all, then we must acknowledge the existence of processes and the need to clearly represent processes. Processes in the real world can only add or subtract, in the sense described above; all other mathematical operations are shorthand notational conveniences. But simply describing the effects of processes is uninteresting: we could take a photograph of the result of a process, or we could otherwise quantify its result (say, by tracking total population in an ecosystem); but these brute facts do not give us any real understanding of what is going on within the process. To represent the internal relations of processes we must use rules. One might say (as Lloyd Morgan did) that the "effective relatedness" of things, both internally and externally, changes in emergent transactions; and the effective relatedness is that which is represented by rules. After defining rules that accurately replicate key features of observed processes, then using Occam's razor and principles of algorithmic complexity we can select our favorites. The goal is to select the simplest (i.e. fewest bytes) set of rules that defines all behavior of interest. In spite of their unpredictability (whether it be inherent or temporary), emergent transactions are lawful and can be studied just like any other transactions. Given that we can describe entities at one level A (e.g. the properties of hydrogen and oxygen) in terms of their probable rules of behavior, and can describe entities at a higher level B (e.g. the properties of water) in the same manner, we are left "only" with the need to define or discover rules that describe how an entity becomes subject to a completely new and different set of rules, i.e. how it becomes a distinct new entity. As we cannot (by the definition of emergence) use any algebra or form of logical inference to determine from one set of facts what another, related set might look like, we can only describe the conditions under which one set of facts becomes another, different set of facts. This requires the use of conditional statements. My work thus far indicates that there is a way to canonically represent rules at a higher level of abstraction than the mere individual rule; I call this approach "Ultra‐Structure" (see Long & Denning, 1994). An operating rule in Ultra‐Structure has one or more factors (IF conditions) and one or more considerations (THEN conditions), which may be thought of as attributes in a relational table. The factors determine whether the rule and its considerations will be inspected and possibly executed. Once a rule has been selected for inspection, the considerations are used to determine in the context of other rules whether that rule will be executed. Abstractly, a rule with, say, three factors A, B, and C, and three considerations X, Y, and Z is a conditional statement which is interpreted as follows: if A and B and C then consider X, Y, Z.
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The factors of a rule are always ANDed together. Logical OR is represented by multiple rules. Each record is unique. Groups of such conditional rules that share the same factors and considerations are called a ruleform. A General Class of Emergence Rules In modeling the U.S. Constitution using Ultra‐Structure, I've found (not surprisingly) a fairly precise set of steps defining how a normal human becomes a U.S. Senator.4 The powers (properties) of Senators are quite different than the qualitative properties of ordinary citizens. For example, Senators cannot be arrested during their attendance at a session of Congress, or while going to or from a session of Congress, except for a treason, felony, or breach of the peace; nor can a Senator be appointed, during the time for which he was elected, to any office which was created or had its pay increased during his term of office; and ‐‐ most importantly ‐‐ their vote counts on matters before the Senate, unlike (say) mine. Likewise, the Constitution specifies the conditions under which a bill may become a law; and we all know that laws have very different properties than bills! With such intentional systems, as opposed to natural systems, we do not "predict" what happens if we add A to B, nor do we explore what happens if we reverse the prescribed procedure for things: we simply collectively, as a society, declare what happens. And in cases of doubt, we have a mechanism ‐‐ the Supreme Court, in the Constitution ‐‐ that is empowered to declare clarifications of rules. . Other examples of intentional emergence are marriage, corporations, and nations. Such emergence‐by‐declaration is obviously a phenomenon that can only occur in cases where institutional facts can be generated by human intentionality. But what about natural emergence? We commonly represent changing states of affairs by modifying the attributes (predicates) possessed by entities. Thus if Tom is a person‐entity and Tom gains 5 pounds, it is still Tom in some fundamental sense, but now his weight attribute is N+5 pounds. If Tom grows a mustache, he is still Tom and we can still represent him by modifying selected attributes. Tom is still the same entity, and subject to the exact same regulative rules as he always was. But suppose Tom turns out to be "swampman", the creature created spontaneously from stuff in a swamp? He still looks and acts like a person (I say), but now he may not be covered by the laws governing treatment of humans, for he is not in fact human. How do we represent this state of affairs? More generally, under what conditions (rules) can we say that entity A ceases to exist and becomes entity B, subject to wholly new and different rules, and having in some cases wholly new attributes? In natural systems, this identity question arises when chemical compounds change their state. We tend to think of ice, water and steam as different states of the same entity; and yet each state acts very differently, i.e. it follows different rules. Thus, from an Ultra‐Structure point of view, liquid water, steam, and ice must be defined and treated as three separate and distinct entities, because each is subject to different classes of rules. The fact that they are inter‐translatable by the addition or subtraction of heat energy is represented by "emergence rules", stating that ice plus heat makes water,
4 -- In modeling the Constitution using Ultra-Structure, I've concluded provisionally that the essence (deep structure) of any regulative legal system is the probabilistic assignment of new legal statuses (new predicates or qualities) to some legal entities by other legal entities.
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and water plus heat makes steam. Thus we may model an entity (e.g. water) plus attributes (e.g. state); but because the resultant acts so differently, we treat certain cases as distinct entities each subject to their own rules. It does not matter whether the new entity is apparent or real; only that it behaves differently. In a strictly descriptive sense, we can imagine a notation having a grammar whereby adding or subtracting Entity X and Entity Y does not follow the classic (resultant) grammar of arithmetic. By definition we cannot specify what any two (or more) entities will add up to; thus the properties that emerge must be declared; they cannot be deduced. And they must be stated contingently, as conditional rules. Furthermore, "adding" cannot be treated as an abstract operation, because in fact what is being added matters: we must define more concrete operations such as "added to X", "added to Y", "added to Z", etc. Each such situation can then be treated as a predicate of an entity: i.e., a status. Thus, in the general case, If Entity X acquires status C, then it (becomes) (is treated henceforth as) New Entity Y. Examples are:
X = bill, C = approved by House, Senate, and President, Y = law
X = ordinary citizen, C = wins election, Y = U.S. Senator
X = U.S. Senator, C = loses election, Y = ordinary citizen
X = hydrogen, C = added to oxygen, Y = water
X = water, C = heat, Y = steam
X = pawn, C = reaches 1st rank, Y = queen. I call these "emergence rules", and the class of such rules I call an "emergence ruleform", for by effectively redefining (renaming) the type of entity one is dealing with in an Ultra‐Structure model, that entity becomes immediately subject to completely new and different rules. This and like facts cannot be expressed in mathematics, which is the notation of resultants; it can only be expressed by a contingent‐rule‐expressing notation such as Ultra‐Structure. In the most general case, for every permutation of statuses that an entity may have, it may be subject to new and different classes of rules. Therefore if its status changes for any reason (including the mere passage of time), it (or the new entities it becomes) may behave in unexpected ways. Perhaps it is for this reason that the general semanticists like to index words: so we don't always presume that an entity at t=1 is the same entity at t=2. It does not matter whether the changes are the result of human intention or not: the application of a rule that declares the existence of new entities whenever existing entities achieve a defined status permits us to replicate, in a computer model, the characteristics that we see in real‐world emergent transactions: they are novel; they are sudden; and they are not predictable by an understanding of their causes. Conclusion We may speculate, like Sellars, that emergence occurs because "at specific degrees of complexity of organization, new properties are formed in order to establish a fresh and simpler point of departure"
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(Blitz, page 180). Although we may not know why any given emergent transaction occurs as it does, "merely" documenting its behavior and rules is still real science, not a yielding to irrationalism5. If at the end we are left to look in wonder at a world where we expect one plus one to equal two, much as we once presumed parallel lines could never intersect, we need not only "Consider and bow the head", as Morgan suggested: we may and indeed must respond with creative new technologies of representation. And then consider and bow the head. References Blitz, D., Emergent Evolution: Qualitative Novelty and the Levels of Reality. Boston: Kluwer Academic Publishers, 1992. Gellert, W., Kustner, H., Hellwich, M., and Kastner, H., The VNR Concise Encyclopedia of Mathematics. New York: Van Nostrand Reinhold Company, 1975. Lewes, G.H., Problems of Life and Mind. London, 1874. Long, J., and Denning, D., "Ultra‐Structure: A Design Theory for Complex Systems and Processes", in Communications of the ACM (in press) Miller, D.L., Emergent Evolution and the Scientific Method. Doctoral dissertation, University of Chicago Department of Philosophy, 1932. Morgan, C.L., Emergent Evolution: The Gifford Lectures. London: Williams and Norgate, 1927. Russell, B., The Analysis of Matter. London: Allen and Unwin, 1927. Quoted in Blitz, D., Emergent Evolution: Qualitative Novelty and the Levels of Reality. Boston: Kluwer Academic Publishers, 1992.
5 -- See D. L. Miller's dissertation
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Emergence Nonlinearity Anomalies of Complex Systems
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Old Goal: Understand the WHY of emergence & nonlinearity New Goal: Accept brute facts. Develop the ability to accurately represent (model) all kinds of addition and subtraction, including transactions that demonstrate apparent emergence and nonlinearity Why is Not the Question
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Attributes Attributes
Attributes
or Quantity or Quantity
or Quantity
Summands
Sum
A B
C
Latin addere, to add derived from ad = to or towards and dere = to put
to put towards Contrast with abdere, to put away
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Emergence:
When an entity is suddenly subject
to a new and different class of rules,
as observed qualitatively or quantitatively.
Operational Definition Includes Nonlinearity & Emergence
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width
height
mass
velocity
acceleration
value color
shape
purpose
age
strength
substance
Entity Attributes
(et cetera)
Distinction Arises from the Aristotelian Legacy
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Entities Statuses
Emergence
Rules
Status AssignmentRules
Entity
Status
Deep Structure of Addition & Subtraction
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Both quantities of entities and properties of entities are additive and subtractive simpliciter:
commutative: a+b = b+a
associative: (a+b)+c = a + (b+c)
monotonic: if a < b, then (a+x) < (b+x)
reversible: if c = a+b, then c‐a = b Resultant Transactions Have Certain Assumptions
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Dimensionless
Dimensional
Dimensional-Plus
ConditionalUnconditional
math
physics
chemistry,
accounting
Science has Known that Not All Things can be Meaningfully Added Together
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ConditionalUnconditional
Low Interaction
Medium Interaction
High Interaction
continuous
chaos
statistics
functions
And Science is Learning How to Represent Complex Interactions... Adapted from Sally Goerner, Chaos and the Evolving Ecological Universe (1994)
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Quantities of entities and properties of entities are not necessarily additive and subtractive simpliciter; most transactions in the real world are:
non‐commutative: a+b b+a
non‐associative: (a+b)+c a + (b+c)
non‐monotonic: if a < b, then (a+x) >, =, or < (b+x)
non‐reversible: if c = a+b, then c‐a b
1 + 1 = 2
1 + 1 = ???
Which are a Special (Limited) Case of the Assumptions for Resultant Transactions
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Factors Considerations
F1 F2 F3 F4 C1 C2 C3 C4 C5 C6
Rules
R1
R2
R3
R4
R5
R6
Universals
A B C D U V W X Y Z
E F G H I J K L M N
But we Still Need to Show Conditions (IFs) and Other Kinds of Considerations (THENs) (Besides Units of Measure)
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Summary We currently create "Numbers‐Plus" by:
adding dimensions to represent qualities
adding extra operational procedures to represent special handling rules But we must also:
explicitly add environmental conditions ("factors") onto the operations
add other "considerations" besides UM that affect how a rule is to be executed. This will permit us to better represent known facts about:
emergent behavior (i.e. properties of sum not predictable from properties of summands)
nonlinear behavior (i.e. quantity of sum not predictable from quantities of summands; output not commensurate with input)
