Chapter 6



In this chapter we further clarify and develop the replicator concept, and we also develop a new concept of generative replication. The primary of generative replication virtue is that it allows us to consider replication of developmental instructions. We relate the replicator concept to complexity, and identify conditions that can lead to increases in complexity.

We emphasize that a replicator is not a thing but an informational mechanism. This is sufficient to counter many of the arguments against replicators. In general, following several other authors, replication involves three conditions, pertaining to causality, similarity and information transfer (see page 119). These establish the broad concept of a replicator.

Next, in section 6.3, inspired by work by John von Neumann, we ask what kinds of replicator and replication have the potential to increase complexity. This leads us to fine-tune the three conditions already established in the literature, and add a fourth, concerning “conditional generative mechanisms” (pages 122-3). When a replicator also satisfies this fourth condition it is called a “generative replicator”. The fourth condition is that within the information contained in the replicator there are instructions that guide the development of its host interactor. In other words, a particular kind of development information is involved.

An example of a biological replicator that is not generative is a prion (as in mad cows’ disease). Genes are clearly generative replicators. We argue that habits and routines can also be generative replicators. We criticize the meme concept to be vague and inadequate in this regard.

We argue that generative replicators have the potential to increase complexity, as long as their information is copied with sufficient fidelity. Crucially, copy error must be minimized. We note that complexity in social systems has increased much more rapidly in the last 10,000 years than in biological systems. This for us is evidence consistent with the existence of generative social replicators.


1. Can the replicator concept survive the criticism to which it has been subjected?
2. What are the weaknesses of the meme concept?
3. What are good examples of generative replication in social systems?
4. How is complexity defined?
5. Why is an examination of the conditions for increasing complexity important?


20 thoughts on “Chapter 6

  1. gmarletto

    Dear Geoff and Thorbjorn,
    I have three queries on replication. Many thanks and best wishes, Gerardo.

    1. Manipulation (Interactor → Replicator)
    I don’t understand why figure 6.1 is not developped from figure 4.2
    The arrows ‘l’ in figure 4.2 may also represent the ability of (social) interactors to manipulate their replicators, thus influencing further replications.
    In more general terms I think that manipulation helps to understand:
    a) why social evolution is faster than the natural
    b) why social evolution gives rise to increased complexity

    2. ? (Interactor → Environment)
    I don’t understand why in both figures 6.1 and 4.2 the ability of (social) interactors to influence their environment is not represented.
    I think that also this ability (how would you name it?) helps to understand the above points a) and b).
    Moreover I think that the consideration of the relation ‘Interactor → Environment’ may help to explain the pervasivity of lock-in phenomena in human history.

    3. Enlistment as replication?
    Maybe I am wrong, but I understand that you think that in social evolution replication is seldom associated to the (pro)creation of new interactors. (The only examples you quote refer to firms: spin-offs, de-mergers, joint-ventures, etc.).
    Diffusion is a relevant example of replication without (pro)creation: a new replicator is transmitted to an already existing interactor.
    May enlistment (enrollment) in a group be considered as another relevant form of (social) replication without (pro)creation? (Where a group may be a political party, a research team, a firm, etc.). Indeed, also in the case of enlistment a replicator is transmitted from one to another interactor (both already existing).
    Note that through the enlistment of new agents a (social) group can increase its ability to influence the environment…

    1. Geoffrey M Hodgson Post author

      Thank you Gerardo for your comments and questions. Referring to each one in turn:

      1. We could have developed figure 6.1 from 4.2, but we chose to give a simplified and different figure. In particular, repicator manipulation could have been introduced in 6.1. But remember that 6.1 covers both biological and social evolution and replicator manipulation is very rare in biological evolution. Replicator manipulation could increase complexity if the host interactor survived longer or produced more offspring. There is no guarantee that this would happen. Even if it did, how could such replicator manipulation be repeated and tested by selection?

      2. Again I point out that figures 4.2 and 6.1 are simplifications. They cannot include every possible causal linkage. I do not deny that manipulation of the environment is important. We chose to focus on the causal linkages that were important for our particular exposition in each case, rather than trying to include everything.

      3. I think that your useful example of organizational enlistment could amount to diffusion by our definitions, if routines were copied.

      I hope this helps.

  2. gmarletto

    Dear Geoff, may we agree on the following?
    1. Manipulation of replicators is relevant in social evolution
    2. Manipulation of the environment is relevant in social evolution
    3. In multilevel cultural selection diffusion and enrollment relate to different levels of replicators. The former refers to the copying of lower level replicators (e.g., technological routines copied from a firm to another); the latter to the copy of higher level replicators (e.g., some innovation routines are shared by a network of firms and research bodies, and are adopted by a new frim enrolled in the network)

  3. Geoffrey M Hodgson Post author

    Yes, except that (in regard to 3) diffusion refers to the copying of a replicator by one interactor from another, where the copied replicator can exist at ANY level within the host interactor. I fully agree with 1 and 2.

  4. gmarletto

    (Still on point 3.) I’m not sure we are using this terminology with the same meaning…
    Going back to my example: the innovative routines of a network are not hosted by its members (firms and research bodies), but by the network itself. In other terms, innovative routines are emergent replicators of the network.
    Other firms and research bodies can share these routines only through ‘enrollment’ in the network.
    The case is different if these innovative routines are copied by another network of firms and research bodies (say, located in another country). I would name this case ‘diffusion’.

  5. reasonableadventurer

    Hi All,

    apologies for my absence, travel and other deadlines have ambushed me.

    Geoff, in your chapter conclusion you note likely presence of a ‘Weismann barrier’, both in terms of biological and/or social contexts. While I understand the logic to your stated position. I wonder to what extent your position is determined by the assumption of relative environmental homogeneity? Tapping into Gerardo’s comments, once the process of niche construction is contemplated by individual firms operating in heterogeneous environments, it is not difficult to see how their organizational routines could be altered by a ‘two-way’ flow of information; regardless of the nature of social culture existing across other firms also assumed to belong to like population.

    Consider a crude example, As a result of lobbying local government to alter be less restrictive on how a firm may operate in its local environment, the nature of organizational routines required to survive (i.e. international market development process vs local exploitation process) may be altered due to the nature of environmental conditions altered by the interacting elements of the firm.

    In this sense, the experiences of the firm, occuring through its interacting elements have brought about a change in the nature of the routines required to survive and thus, subject to ongoing replication.

    Without not wanting to start another debate of the merits of including Lamarckism in our approach to organizations, is it possible that holding onto the assumed presence of some form of Weismann barrier is holding back our collective abilities to contemplate what habits / routines are subject to replication across all organizations ranging from 1 person firms to those with 1000s of employees?

  6. Geoffrey M Hodgson Post author

    Welcome back, Reasonable Adventurer. In the book we point out the advantages and possible evolution of a “Social Weismann Barrier”. This would mean that the capacity of changing environmental conditions (or other developments) to affect the information encoded in social replicators (habit, customs or routines) would be limited. Our argument does not depend on a constant environment through space or time. On the contrary, the case for a Social Weismann Barrier is reinforced by changing environments and environmental non-homogeneity.

    It is important to distinguish here between the possible adaptive advantages of a single entitly, and the possible adaptive advantages through time in a changing population of entities. Your example concerned a single entity, adapting to circumstances. For a single entitly, barriers to adaptation are often disadvantageous. But if social entities completely change the information in their habits or routines in response to every enviroronmental change, then the population as a whole does not retain important information relevant to survival in past circumstances that may recur. It is often vital that such information is retained: even if it is not used by one entity, its replication may help others survive in future circumstances.

    As a consequence of this adaptation-retention trade-off, successful species (like humans) often develop higher-level habits and routines that have more of a multi-purpose character and help individuals adapt to circumstances under changing conditions. Boyd, Richerson and others have argued that humans developed a sophisticated language and culture partly in response to rapidly changing climatic conditions over the last 100,000 years. These higher-level habits and routines did not mean that the entire vocabulary of a language changed as people moved from hot to freezing conditions, or back again. The destruction of the preceding vocabulary would have destroyed much information that was valuable in current or future ciurcumstances. The rules of language change relatively slowly (compared to say modern technology or fashions). Their inertia is an example of a Social Weismann Barrier.

  7. len wallast

    Good morning Geoff and Thorbjørn,
    Please notice that there are two contradictory definitions of entropy in the literature: 1) the (original) thermodynamic definition; 2) Shannon’s definition. The two definitions only differ in sign (I neglect Boltzmann’s constant).
    All of this sign-business regarding entropy has caused and still causes a lot of confusion in the literature. Are you aware that you use the thermodynamic entropy concept e.g. on page 121, line 20, on page 127, line 10-11 and 20; and you use the Shannon entropy concept e.g. on page 127, line 24 and on page 128, lines 19-30 and formula (6.1)? My suggestion is to stick to the Shannon definition like I do because positive entropy (= information) is much easier to deal with and prevents confusion.

    On most of the subjects of this chapter I have commented earlier and also in my book on evolvodynamics. I owe you to express that the material on your page 128 has stimulated me very much to reformulate input and output selection in (what I think to be) the proper mathematical way.

    I welcome further your idea that an interactor hosts replicators. I consider interactors (sectors, states) to store a (very large) stock of replicators.

    As to the remarks of gmarletto and reasonableadventurer regarding the role of the environment and as far as I understand their sophisticated non-mathematical prose correctly, don’t you think that their considerations, like the mathematics also suggests, might be the reason why we should deal with the environment as another interactor (with input and output) in the stead of as a selective condition only?
    In this regard: what do you mean by the text of the first nine lines of page 127? Do you perhaps support the idea that the evolving individuals form the exclusive output and that the environment is the exclusive input that feeds the evolving individuals? I think this is too narrow. There is also output of the environment and input of the evolving individuals (e.g. labor). Both inputs (of environment and individuals) are necessary to produce the output of the environment (think of farmers cultivating their land) as well as to procreate evolving individuals as output. The consequence is that there are at least four different (i.e. more than only a single one) conditional entropy outputs given one of the two entropy inputs. Moreover there are also at least four different conditional inputs given one of the two entropy outputs. All of that is related to the statistical dependence of output and input without which entropy growth and evolution cannot subsist.
    All the best, Len

  8. Geoffrey M Hodgson Post author

    Good morning to you Len. Your point about entropy definitions requires reflection on our part, Thanks very much for that.

    Regarding your question concerning the first nine lines on page 127, I do not think that we intended the narrower meaning. Again you offer useful food for thought.

    Regarding whether the environment is an interactor or not, let’s postpone that until we discuss the definition of an intercator in the next chapter.

  9. Jerry Ulman

    I wish to offer an alterative to habit in accounting for generative replication in social systems, one that is more parsimonious and based on the empirical research of an established natural science of behavior. In the glossary, habit is described as “a disposition to engage in previously adopted or acquired behavior (including patterns of thought) that is triggered by an appropriate stimulus or context.” There are three kinds of habits: corporeal habits that replicate through behavioral imitation; linguistic habits that depend upon language for their replication; and habits of thought that include linguistic habits and culturally acquired emotional habits.
    First, lets examine the disposition concept. It is an unnecessary complication; an obsolete hypothetical construct originating from nineteenth century philosophers and theorists (e.g., Thorstein Veblen). Not uncommonly, situations lacking apparent contiguous causal events invoke constructs asserting the existence of reified entities, disposition being a prime example, Also, note that disposition is an antecedent causal variable, not a selectionistic process
    Next, consider the problems with habits. From the behaviorological perspective (please see my Chapter 3 posting), habits are—in effect—behavioral repertoires. A repertoire refers to all the behaviors an individual can perform; or specifically, the set of actions relevant to a particular setting or task. A storage metaphor is not needed. Contingencies of reinforcement change behavior, and in doing so the organism is changed. For example, when we learn to keyboard, that skill is not stored someplace; we have acquired a skill. Operant behavior (in contrast to respondent or reflexive responses) isn’t triggered; it is evoked by a discriminative stimulus, a stimulus in the presence of which some action has been repeatedly reinforced. When I pick up my dog’s leash she immediate comes to the back door. We are then on our way for a nice long walk. The jingle of the leash has become a discriminative stimulus because of her history of reinforcement. Referring to a trigger does not convey how behavior comes under control of the antecedent stimulus, whereas the term discriminative stimulus does do that.
    What about the different kinds of habits? Understood behaviorologically, lingual habits could be replaced with verbally-governed behavior, actions shaped and maintained by a particular cultural milieu. And corporeal habits could be replaced with event-governed behavior, behavior controlled by nonverbal stimuli. Both kinds of behavior can be acquired through modeling as well as by direct training or simply from chance circumstances.
    What, then, is verbal behavior? It is behavior that has been mediated by other actions. For instance, a speaker gives an instruction and a listener follows the instruction; following the instruction mediates the verbal behavior of speaker. Verbal behavior is shaped and sustained by a verbal environment; that is, by people who respond to behavior in certain ways because of the practices of the group of which they are members. With written language, verbal behavior become considerably more complex. But the point here is that the operant concepts—verbally-governed behavior and event-governed behavior—could completely replace the habits concepts. Notwithstanding Chomsky’s ridiculously unsound review of Skinner’s book, Verbal Behavior; if there is any doubt about the viability of this work, survey the wealth of experimental and conceptual research reported in the journal, The Analysis of Verbal Behavior—accessible online:

  10. Geoffrey M Hodgson Post author

    Thanks Jerry for your thoughts on habits. The fact that a concept is “nineteenth century” does not make it wrong. Consider Darwinism, for example. Actually, the concept of habit goes back at least to Aristotle. He distinguished between two meanings of the word: as a disposition and as a behavior. He further argued that the essence of something cannot simply lie in its behavior, because when the behavior (temporarily) ceases, then the behavioral description would be invalid. But a postman is still a postman even when he does not deliver letters. We define a postman as a person with the capacity to occupy and act in a particular social role. If we define him as someone posting letters, then he ceases to be a postman when he takes lunch.

    Darwin confused the two meanings of habit. They were disentangled by William James in his 1890 classic on psychology (very late in that century). But his approach was soon eclipsed by the rapid rise of behaviorist psychology from about 1920. For behaviorists, behavior is everything, and their approach dominated psychology until about 1970. It still has its adherents – Jerry seems to be one, if I am not mistaken. By contrast, some psychologists adopt a concept of habit that is similar to ours in Darwin’s Conjecture. For example, psychologists David T. Neal,, Wendy Wood, and Jeffrey M. Quinn wrote in Current Directions in Psychological Science (2006) “Habits are response dispositions that are activated automatically by the context cues that co-occurred with responses during past performance.”

    We stick with a similar dispositional definition of habit. We also retain the storage metaphor. When we learn something new neural connections occur in our brains. Basically, these store information, at least in the Shannon-Weaver sense. We know that there is much more to information and knowledge than that, particularly the contextual nature of knowledge. It is “embodied” or “situated”, as some psychologists put it. It is here that the storage metaphor can be misleading. All metaphors have to be handled with care.

  11. len wallast

    Perhaps I may contribute a little to clarify the confusion about replicators, habits, dispositions and behavior.
    Understanding selection requires that we realize that selection is a dynamic process and also that the results of the selection process (the two interactors, which consist of the population and its environment), that have evolved from the selection processes in the course of time, are dynamically changing as well.
    Selection is concerned with events that occur in the course of time t when drawing samples (replicators) from the content of a “box”. That box cannot be anything else than a stock of replicators because else we would not draw that kind of samples from it. We cannot select replicators from something else than a box of replicators. Further, as the contents of the box varies with time (because evolution is a nonstationary stochastic process of accumulation and withdrawal), the conditions of selection stay only unchanged if selection is restricted to a narrow time-interval (t,t+dt) of infinitesimally small time-length dt at time t. This is the way we must study the selection process in order to keep it tractable. The consequence is that evolutionary selection is a process of dynamically changing flows of information, whereas the “box”, from or into which is selected, stores dynamically changing stocks of information. By selection during (t,t+dt) we gather only a very small vanishing portion of the contents of the box, while the stock of the box remains constant to the first order of dt during (t,t+dt).
    Thus we must make a sharp distinction of what is in the selected flow during (t,t+dt) and what is in the box at time t or at time t+dt. The selected flow of replicators consists of events: actions during the time-interval (t,t+dt). What remains unselected during (t,t+dt) remains in the box during that interval of time and cannot reflect any really occurring event. The replicators that remain in the box are not selected and represent the “frozen” stock-situation at time t+dt.
    Can we call the replicators in the box habits? Can we call them dispositions? Can we call them behavior?
    Similarly, can we call the selected replicators habits? Can we call the selected replicators dispositions? Can we call the selected replicators behavior?
    The answers to these questions depend on the definitions we attach to these concepts. I am inclined to call the replicators in the box dispositions because a disposition should not reflect an event occurring on (t,t+dt), but rather a stock of available knowledge, experience, expertise and understanding that is slumbering until it is selected and actively engaged in the future after time t+dt.
    Once a replicator is selected from the box, it is no longer an inactive disposition but an actively engaged replicator in the process of selection. Thus in line with Jerry Ulman we can then indeed state that a disposition is not a selectionist item. But clearly, this is only a matter of linguistic definition. If we adhere to identify a replicator with a disposition irrespective of whether it is involved in selection at the moment or not, then a replicator is always a disposition.
    If the term habit or the term behavior is further used to indicate the active involvement of a selected replicator during (t,t+dt), that will also entirely be a matter of linguistic definitions. H&K choose to identify the unit of selection (the replicator) with habits and routines. As I see it, this is a good manner of getting acquainted with the mathematical selection mechanisms. But it is not necessary. In the end we will find out that our units of selection form a collection of binary sequences that maps one to one unto the set of various habits and routines we insisted to discern originally.

    A final note: I have avoided in the above to discuss the role of bitpulses as a preferred mathematical substitute for replicators. There is a manner to bridge the gap between bitpulses and replicators as units of selection, so that what I said here is still applicable despite that the bitpulse is the mathematically preferable unit of selection.

  12. Geoffrey M Hodgson Post author

    Len writes: “Selection is concerned with events that occur in the course of time t when drawing samples (replicators) from the content of a ‘box’. That box cannot be anything else than a stock of replicators because else we would not draw that kind of samples from it.”

    This is not the way we see selection. We make a distinction between objects (“selection of”) and outcomes (“selection for”) of selection. Replicators, in our view, are not THINGS in a “box”. Replicators are program-like bits of information that are hosted by interactors. The interactors would be the entities in the “box”. When subset selection occurs (i.e, some entities are removed from the box) the interactors are the objects of selection. An outcome is a different pool of bits of replicator-information hosted by the surviving interactors.

    Len’s usage of these key terms differs in these important respects from ours.

  13. len wallast

    Dear Geoff and Thorbjørn,
    I hoped to bridge the gap between your replicator concept and bitpulse selection. Unfortunately in vain.
    You do talk about interactors that are now entities in the “box”. Well I agree with that provided we see the “box” as a stock containing all the interactors that are objects (states) of selection.
    But do you realize that we can only select an infinitesimally small portion of the content of the box during the time-interval (t,t+dt) of selection? That is, there is no time to select all the objects (the interactors) during (t,t+dt) like you won’t have the time (even if you have the money) to buy today all the cars that the economy has in stock at the moment. Clearly, if you choose the interactors as the objects of selection, they must be the smallest units that can independently be selected. However, it appears that your interactor is a stock variable and unfortunately a very big one. How can I select the complete replicator stock hosted by that interactor at once during (t,t+dt)? That would be a miracle. I don’t suppose that you think that the complete time-range since the big bang until our present day is available to clear that huge selection job. If that would be what you mean, how are we going to repeat that selection job tomorrow?
    There is of course an escape by asserting that the selection of an interactor is done slowly bit by bit e.g. by replacing replicator R1 by replicator R2 on (t,t+dt), and R4 by R3 on (t+dt,t+2dt). However, that implies that we are not selecting interactor-wise but bit-wise. And as you know, the bit is the smallest unit of information there is. Hence this confirms what you try to refute in vain: “At the basis selection is concerned with the selection of bitpulses”.
    You really need a much smaller object of selection than an interactor.
    Regards, Len

  14. Geoffrey M Hodgson Post author

    Len, I appreciate your good intentions. But let’s try and be clear. First I suggest we get rid of the “box” image. It’s not particularly useful. Second, remember our ontological commitments in the earlier chapters. We wrote of “complex population systems”. The “populations” here are of interactors. Selection can be in one of two forms – subset selection or (more broadly) successor selection.

    You write “You really need a much smaller object of selection than an interactor.” Where do we refer to the size of the interactors – the entiities in a population? Nowhere do we say that they have to be big. Amoeba are interactors. So too are bugs and insects. You refer to an interactor as a “stock variable”. Nowhere do we say this. Instead, interactors are the members of the population in a population ontology.

    Your concepts of replicator and interactor do not seem to be ours. So the conclusions you draw are not implicated by our definitions. Please refer to them. Your argument seems to be based on different understandings of these concepts. What are your definitions of a replicator and an interactor?

  15. len wallast

    Good morning Geoff,
    You missed the point I made. Perhaps I have not been clear enough, but it is not an easy point to make in a few sentences. Perhaps the space to elucidate it was neither enough. I will try to state my arguments here somewhat differently.
    But FIRST: I do not see the necessity to express myself in accordance with the ideas, terminology and concepts of your selection theory or to adopt these beforehand. The way I choose my arguments is a personal matter that I wish to decide about myself. And I have the right to base me on results I have derived elsewhere. We have a different view about the details of the Darwinian selection process. Ok, but let it not disturb what we share: Generalized Darwinian evolution. As long as there is controversy about the precise interpretation of the selection process it would be rather strange to restrict myself to only explaining your model and to accept it uncritically. Science needs the mutual exchange of arguments.

    SECOND: Don’t think that I do not appreciate the contents of your and Thorbjørn’s book. I consider it as a landmark in economic and social theory formation noting the depth of argumentation, the width of application and the bridges it offers for others working in this field of exceptional scientific interest. As far as it applies to economic applications, the book is a real relief after so many years of the one-sided doctrines expressed in orthodox theories of economic behavior. Furthermore, the book propagates a very ambitious claim: the universal character of Darwian selection including the complete social domain of behavior. That is a message that stands out, but its high ambition makes it also quite vulnerable with respect to criticism and dispute. We should not forget that we are still in the research phase of Generalized Darwinism.

    THIRD: The subject I was writing about: I was of course not referring to the size of bugs and insects, but to the size (the entropy content) of an interactor relative to the entropy content of a single replicator hosted by that interactor. I concluded from your explanations that in your model an interactor hosts a lot of replicators on the average. Furthermore I was also referring to the TIME (the order of events) required to execute selection. Time is a very much maltreated concept in economic theorizing and we must be very cautious not to fall into the same trap as economic orthodoxy does. We really need to take into account the role of time in the mathematics of selection in order to avoid inconsistencies, certainly when we are explaining selection.

    You discern between subset selection and successor selection. It is my impression that your definitions of subset selection and successor selection are quite from complete especially with respect to the many questions left that appear on the road when one tries to conceptualize the concepts and the mechanisms involved with it in a general system of mathematical equations. It appears to me that mathematical conceptualization of the theory is still quite undetermined notwithstanding the given outlines of your selection theory. And ultimately in economics the test of mathematics will decide the matter.
    As far as I understand it, your subset selection is the shakeout of interactors without replacement. I quote from your page 94: “Successor selection involves replication, whereas subset selection is a simple elimination process.” Your use of the term subset is perhaps based on the idea that the removed interactors form a subset of the entire population.
    On the other hand, your successor selection is concerned with the selection of a novel individual (in a population ontology). I quote from page 99: “In nature, successor selection often occurs in cycles that correspond to the replacement of one generation by the next”. What I infer from this now is that your successor selection is more or less concerned with the output selection of an interactor. Your choice of the term “successor” is probably based on the idea that a new generation of interactors is selected. As you suggest, this may be achieved by replication by which you appear to understand that the replicator repertoire of the newly created interactor is a copy of an existing repertoire of replicators. But successor selection can also involve the introduction of new variations of the replicators. I take it that you add this part of the selection model to explain entropy growth (e.g. volume growth and/or the growth of complexity of the interactors).
    Subset selection might thus be called a form of input selection of interactors, because entropy input is sacrificed. Successor selection might then be called a form of output selection of interactors, because entropy output is produced.
    It is not very clear to me whether you also consider the injection of a new variation of a single replicator in an existing interactor as a form of successor selection (relevant for non-biological selection), but the answer to this question is of little significance relative to the problems with respect to that of insufficient selection time.

    What I said before, concerning the latter subject, was that there is insufficient time to select an interactor on an infinitesimally small time-interval of selection. This assertion is based on the fact that evolutionary processes are nonstationary. This implies that statistical time averages of selection over a finite time-interval of selection do not exist. Only statistical averages of selection at a particular fixed time exist These averages are averages of the sample selection space at that particular time. Therefore a sample space of selection (a “box”) cannot exist if selection is stretched out over a finite time of selection. In order yet to deal with meaningful statistical time averages of selection and a meaningful mathematical description of the selection process we must therefore restrict selection to an infinitesimally small time-interval (t,t+dt) of selection of vanishing time-length dt. This warrants that the selection probabilities do not change to first order of dt in the process of selection.
    What then is the problem when we are going to select interactors stretched out over a finite time-range of selection, or stated in your terminology: what happens if successor selection occurs in cycles that correspond to the replacement of one generation by the next. Well, there are no meaningful selection probabilities that can be defined over the finite cycle time of these generations, because evolutionary systems are nonstationary. At each particular time t momentary selection probabilities are different. Thus, there is no possibility to define and gauge the probabilities by which the interactors are selected. Neither is it possible to define what sample space is involved with the selection of the interactors because (in mathematical terms) the set of selectable interactors does not form a workable partition. Interactor selection is therefore completely impossible. When we can’t know the probabilities of selection of the interactors, it is also entirely impossible to know their entropy content. This disqualifies any attempt to quantify your selection theory by a valid mathematical system of equations. Your selection theory may only work for stationary evolution, because only then selection probabilities remain constant. This is e.g. an economy where all flows and stocks remain the same in the course of time: a boring economy that keeps replicating itself with zero real growth. Is this what we wish to explain?

    In order to formulate a workable theory of selection, one must accept that selection must be restricted to an infinitesimally small time-interval (t,t+dt) of selection. There is no sense in selecting complete interactors on such a short time-interval, because the time-duration dt of selection must be chosen so small that nothing really happens at the finite scale of interactor selection. (Note it is here where we encounter the monumental size difference between an interactor and a replicator). The only mathematically correct solution is to restrict selection to the selection of infinitesimally small bitpulses. In a population ontology these bitpulses can be in two different states (population and environment) with determinable probabilities of selection at a particular time t. Your selection model can perhaps be repaired if selection is restricted to infinitesimally small replicators on (t,t+dt). That is the assistance I offered before. In conclusion, my contribution is only meant to be of assistance in getting a truly generalized theory of Darwinian selection. I would appreciate if you take it that way.

    The flash of the above idea of differential selection on an infinitesimally small time-interval (t,t+dt) occurred to me in the year 2006. Before that moment, I had spent many many nights, weeks, months and in the end many, many years in vain trying to incorporate Shannon’s stationary selection theory of information transference in a general evolutionary framework. The problem was: to bridge the mathematical gap between Shannon’s stationary selection theory and the nonstationary stochastic nature of evolutionary processes. Before, I had literally and desperately tried every other possibility I could think of. The many years it took me to find the clue, might indicate that the idea is not self-evident at all. I expect therefore that the above explanation is not enough either to understand the impact fully. Thus I must further refer to my book, where the subject is explained in much more detail (See for instance the Shannon-Darwin time-compression transition).
    Regards, Len

  16. Geoffrey M Hodgson Post author

    Len, thanks for your extended thoughts on this. Here a just a two very brief and selective responses, to a small part of what you say.

    First, of course you are not obliged to adopt our definitions. But if you use the terms “interactor” or “replicator” in a way different from us, then unless you supply clear alternative definitions then your meaning cannot be understood. I am still unclear what YOU mean by these terms.

    Second, although the terms “subset selection” and “successor selection” are ours, the distinction is taken from George Price’s widely-used formalization of the selection concept.


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