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A Formal Framework for Social Learning in Innovation Communities Russell Cameron Thomas
Department of Computational Social Science Krasnow Institute of Advanced Study George Mason University Fairfax, VA 22030 Email: http://
[email protected]
Abstract This report proposes a formal version of Boisot’s Information Space (I-space) framework and Social Learning Cycle (SLC) theory to support research in social learning in innovation communities, broadly defined. The benefit of a formal framework is that it will enable more systematic empirical research and also computer modeling.
I. I NTRODUCTION One of the most vexing problems in innovation research is to understand how disparate knowledge, expertise, and information come together in a social setting to yield game-changing, breakthrough innovations (hereafter labeled “macroinventions”). Macroinventions are not important merely because they are ‘better’, ‘new’, ‘surprising’, or even ‘popular’. What makes macroinventions important is that they have a restructuring effect at large on social, economic, technological, and/or political landscapes. While there are viable research methods for studying microinventions (i.e. incremental improvements and recombinations), there are few viable methods to embrace the richness and complexity of macroinvention processes. In his analysis of technology evolution and macroinvention, Mokyr [1] argues that technology evolution is fundamentally an epistemological phenomenon in a social context – i.e. social learning. Therefore it is necessary to focus on how knowledge is created or acquired, transformed, and ultimately put Final Report for OR944: The Process of Discovery and Its Enhancement in Engineering Applications (Spring ‘12)
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to use by individuals, groups, and larger collectives. Moreover, the research focus needs to be on social processes, social forces, and social effects rather than individual behavior, incentives, and imagination. Mokyr provides a good summary of the research challenge: “Macroinventions are seeds sown by individual inventors in a social soil. There is no presumption that the flow of macroinventions is the same at all times and places...The environment into which these sees are sown, is of course, the main determinant of whether they will sprout. There are no one-line explanations here, no simple theorems. It is hard to think of conditions that would either be necessary or sufficient for a high level of technological creativity. A variety of social, economic, and political factors enter into the equation to create a favorable climate for technological progress.” ( [1] p 313) Therefore, in order perform research aimed at uncovering the “necessary and sufficient conditions” for macroinventions, researchers need a framework that encompasses a wide range of phenomena and processes involved in social learning. For example, researchers would like to study the interplay between individuals, social networks, institutions, and formal organizations to understand what conditions promote or inhibit macroinnovation. Researchers would like to understand why macroinventions sometimes cluster in time and in space, and what sorts of interventions might improve the chances of achieving macroinventions in socially-important areas such as cyber security, clean energy, and others. Another hot research topic is on how innovation might be organized and governed in new ways, e.g. open innovation communities. One promising framework is the Information Space (I-space) proposed by Max Boisot [2], [3]. Using the I-space framework, Boisot proposed a theory of social learning associated with invention and innovation called the Social Learning Cycle (SLC). While both the I-space framework and SLC theory are compelling, they have not had much effect on the research community as yet, partly because the I-space framework is qualitative and suggestive. This makes it hard to apply to empirical or theoretical research in a way that would yield important results and attract attention in the research community. Specifically, it is not possible to code empirical data in the I-space in a way to test hypotheses and computer models of social learning using I-space lack both empirical grounding and theoretical precision.
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The primary contribution of this report is a formalization1 of the three dimensions of the I-space framework—Abstraction, Codification, and Diffusion. An ordering procedure is defined for each dimension which can be applied to knowledge artifacts and processes to specify their location in the I-space. Furthermore, formal definitions are given for characteristics of the Ispace, including ’path’, ’trajectory’, ’similarity’, ’accessibility’, and ’proximity’ (an analog to distance). The semiotic theories of C.S. Peirce are used to formalize the dimension of Abstraction. Shannon’s information theory and computational theories of language complexity are used to formalize the dimension of Codification. Finally, social structuration theory is used to formalize the dimension of Diffusion. Before proceeding, a brief clarification is warranted on the word ‘knowledge’ as used in this report. ‘Knowledge’ in this report refers to pragmatic knowledge – the ’how’, ’what’, ’who’, and ’why’ associated with practical goals and aims – rather than the more generalized objective ideal of “justified true belief”. Any types of knowledge can be pragmatic, including tacit, conceptual, theoretical, definitional, etc. But it is not “truth for truth’s sake”. As such, pragmatic knowledge may or may not be fully justified. It may be provisional, sketchy, or even mis-guided. It is always subject to refinement, revision, and even rejection and replacement. What counts is that it is relevant and deemed useful by agents and communities engaged in the tasks of pursuing those practical aims. Also, pragmatic knowledge is both situated and socially constructed, even when it strives for objective universals. Finally, just as light is defined relative to darkness, what constitutes pragmatic knowledge in every situation is inevitably bound to its opposites: uncertainty, incompleteness, ignorance, ambiguity, the incommensurate, the unknowable, the undecidable, and the infeasible realms. Thus, pragmatic knowledge at any time represents a compromise between what is accessible to the mind(s) and what is out of reach, and the effort and cost of extending that reach relative to the perceived benefits. 1
For pragmatic reasons, Boisot himself was not an advocate of formalizing I-space by quantifying the dimensions: [3] p 64:
“To the extent that knowledge assets can be made distinct and identifiable, they can be in principle located in I-space. Although the exercise can be carried out quite systematically, in my experience managers prefer to be intuitive and to offer intuitive ‘eyeballing’ estimates of their location. Using simple scales they are usually able to rate with out much difficulty the degree of codification, abstraction, and diffusion of their firm’s products, technologies, and organization elements. They first do this individually, reaching consensus iteratively through discussion.” On p 65, he provided a “Scaling Guide” which can assist in rough positioning in the I-space in three ordinal categories for each dimension, labled “High”, “Medium”, and “Low”.
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The report is organized as follows. Section II provides an overview of the I-space framework and SLC theory and explains why they are relevant to research on macroinnovation. Section III describes the proposed formalization procedures. Conclusions and future research directions are provided in Section IV. II. B OISOT ’ S F RAMEWORK FOR S OCIAL L EARNING In two seminal works [2], [3], British management scientist Max Boist proposed a framework and theory for social learning. He defines ‘social learning’ as the methods and processes involving knowledge for practical purposes– i.e. creating, acquiring, transforming, and ultimately utilizing knowledge by individuals, groups, and larger collectives. Boisot’s explicit aim was to unify many different theories associated with social learning, from disciplines as diverse as sociology, anthropology, psychology, management and organization science, political science, and cognitive science. His approach was to define a conceptual space for possible phenomena, then to use that space to map extant or hypothesized phenomena in such a way to formulate a theory of social learning. Boisot’s primary phenomena of interest are artifacts used in communication, learning, discovery, or knowledge utilization. In [3], he calls them “knowledge assets”, but I will use the more general term “knowledge artifacts”, because they may or may not be viewed as assets by subjects in the system. A knowledge artifact can be a thing (i.e. a message, a book, a tool, a design, etc.) or a realizable process (i.e. a training process, a production process, a communication process, and information processing process, a utilization process, etc.). Thus, it is through knowledge artifacts that people create, transform, and use knowledge for practical aims. This is not meant to reify knowledge. Instead, knowledge artifacts can be seen as the tangible, observable instantiations of knowledge, much like a circle drawn on a sheet of paper is an instantiation of the Platonic idea of ‘circle’. In Peirce’s semiotics, knowledge artifacts can be seen as signs of knowledge, and are thereby used by people in because of their signifying capabilities. Boisot’s primary behavior of interest is the transformation of knowledge artifacts through lifecycle of discovery, learning, and diffusion. Specifically, he’s interested in tracking changes in the form and informational characteristics of knowledge rather than the domain-specific details. This abstract approach is an understandable consequence of his aim of unifying disparate theories of social learning, but it has an unfortunate side effect of making it harder to put his framework
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into practice in research. The next two subsections will describe Boisot’s Information Space framework and the Social Learning Cycle theory. A. Information Space Information Space (‘I-space’) is a conceptual space of possible knowledge artifacts that is framed by three dimensions: 1) Codification, 2) Abstraction, and 3) Diffusion. The primary purpose of I-space visualize knowledge discovery, creation, transformation, diffusion, and utilization in a social context via locations and trajectories through the I-space.
Codified
News
Abstract
Abstraction Concrete
Common Sense
Uncodified
Personal Knowledge
Textbook Knowledge
Codification
Proprietary Knowledge
Personal Experience
Undiffused
Fig. 1.
Diffusion
Diffused
Information Space (I-space) with six sample knowledge artifacts plotted at their approximate position in I-space.
(Source: Author, after Boisot [3])
Boisot’s fundamental insight behind I-space comes from the pragmatics of discovery and learning, especially the economics of attention, communication, and learning. Boisot starts with these pragmatic principles: •
It is costly to communicate and absorb new information and therefore people will economize when possible.
•
The two primary ways people economize on information/knowledge flows is to 1) generalize or 2) abbreviate or compress the information.
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•
In the act of economizing, you gain something (generality and/or compressed expression) but you also lose something (richness of detail, specificity, context, concreteness). Many times you don’t want to lose the detail and concreteness.
•
Knowledge can both be ’free floating’ in the form of information and it can be also be ’embedded’ in products and services where the knowledge is revealed in the course of using those products and services.
•
What is known, how it is communicated, and how it is known/learned/used changes with social context – i.e. between personal, group, and collective – in ways that depend on the nature of those contexts.
From these principles he defines the three dimensions that are described in the following numbered sections. To make these dimensions meaningful, it is necessary to first define a single epistemological domain and a single social domain. While I-space is generalizable to nearly any epistemological and social domains, it only makes sense to plot points in I-space when a single epistemological domain and a single social domain is specified in advance. Later in the paper I will discuss how to handle the interaction of multiple domains. Also, each dimension is characterized within an interval from ‘least possible’ or ‘most possible’. Therefore the I-space is bound in all dimensions though it might have infinitely many points within those bounds. 1) Codification Dimension: This dimension draws distinctions according to the degree to which information associated with the knowledge artifact is explicitly encoded or not (i.e. tacit). The higher degree of codification, the more the information can be communicated in a compressed (i.e. encoded) way. Conversely, the less codification, the more the information is carried in the direct, first-person experience and context. While it may be ideal from an economy of communication viewpoint to encode all information as much as possible, it may also be infeasible or undesirable. Consider the simple example of a major league baseball game of some importance, say the seventh and deciding game in the World Series. How could a fan learn about the game? Perhaps the most highly coded way of capturing what happened in the game would be through the final score for two teams along with their names or simply initials. To someone who knows US major league baseball well, this would certainly communicate the outcome of the game and the Series. For some purposes, this would be useful and sufficient. However, if all baseball fans felt this way, then no one would watch games, let alone attend games in person with the associated costs.
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Box scores and player statistics is another highly encoded way of communicating the outcome, but requires more context and knowledge on the part of the receiver. Play-by-play descriptions such as a transcript of a radio broadcast would give much more concrete information and value, but only to someone who can visualize what is going on in the field of play and how the descriptions relate to the rules of play and determination of the winner. It is somewhat encoded but not very condensed compared to the two previous examples. Continuing in this direction, watching the game on TV or watching the game in the ballpark give much more vivid, rich, contextual information but at the expense of time and attention to take it all in. Perhaps least codified form of information is the first-person experience of players on the field. While sports journalists are constantly asking players questions like: ”What were you feeling...? What were you experiencing...?”, the verbal response of ball players is a faint and distant reflection of their first-person experience, which may ultimately be ineffable. 2) Abstraction Dimension: This dimension draws distinctions according to the degree to which information associated with a knowledge artifact is either concrete/specific or general/abstract. The higher the degree of abstraction, the more can be communicated with less, separating it from any specific context. Conversely, with less abstraction, the information provides insight to a specific context or state of the world. While we often want to communicate or think in the most general terms, it may be infeasible or undesirable or both. Consider another example from baseball. At the most concrete level, we might communicate information about a specific pitch at a specific point in the game. We could register it’s starting and ending velocity, spin rate and axis, trajectory, and even the sound of the ball passing the batter and hitting the catcher’s glove. We could even enumerate the detailed properties and measurements of that specific baseball. That would be a very concrete set of information and would require a lot of effort to communicate and also to take in. A more general way of communicating would be to use expressions common to play-by-play announcers, e.g. ”a curve ball, low and away”. Of course, this describes a class, not a single instance, but for many purposes it conveys the essential information while leaving out extraneous details. At a more general level still, the pitch could be describe via parameters in physics equations that describe the laws of motion for baseballs. The equations are much more abstract and general because they describe all possible pitches and trajectories, not just those of a single class. In those equations might be hidden types of pitches that no one has yet discovered or perfected. Yet in this most abstract
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level of information, the receiver will probably not be able to understand much about how that particular pitch worked against that particular batter in that situation. Just because something is possible or conceivable in the abstract doesn’t mean it will work out as intended in practice. 3) Diffusion Dimension: This dimension draws distinctions according to the degree to which information associated with a knowledge artifact is either accessible to or used by an individual, a group, or larger collective. The higher the degree of diffusion, the more the knowledge has a collective effect or is processed collectively. Conversely, with less diffusion, the knowledge is only accessible or utilized by a single individual or small group. This is a somewhat different meaning of ‘diffusion’ than is common in the innovation literature, where ’diffusion’ simply means “widely spread”. In Boisot’s I-space framework, it is conceivable that information or knowledge might be held by 100% of a population but still only be accessible and usable on an individual basis, with only individual benefits. This would be coded as ’low diffusion’ in I-space. Like the previous two dimensions, there are trade-offs in the Diffusion dimension. To achieve large scale benefits, we may want some knowledge to be maximally diffused in a community. But in emphasizing collective utilization and benefits we would miss the specific, situated experiences and learning of small groups and individuals. Back to the baseball example, let us consider the fans attending live at the stadium as a community. Detailed information about the experience of a particular fan in seat G12 would be at the lowest point on the Diffusion dimension. That fan’s experience depends critically on who they are (home team fan? visiting team fan? not a baseball fan at all? under the age of 3?) and their immediate circumstances (do they have a good view? are they too warm or too cold?). Conversely, at the highest point on the Diffusion dimension you might identify information that only applies to the crowd as a whole, such as their ‘spirit’ and ‘energy’, or lack there of. Another example of collective phenomena is the crowd-generated ‘stadium wave’. This general information may be sufficient to describe the general state of the crowd and it’s collective properties, it is not adequate to describe the specific experiences of any individual fan or group. 4) Statics and Dynamics in I-space: With these three dimensions defined, it is possible to use I-space to perform static analysis of knowledge artifacts and actors involved in innovation. Boisot generally represents knowledge artifacts as points in the I-space, presumably with a unique coordinate in each dimension. Actors, on the other hand, are often represented as regions or collections of points in the I-space, which seems to express the capabilities of each actor (e.g.
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person, group, organization, institution, etc.). As a tool for static analysis, I-space doesn’t show any particular benefits over a simpler form such as a text table. The real power and benefits of I-space as a research tool come when it is used for dynamic analysis, that is the flow of knowledge artifacts and information throughout the lifecycle of innovation. Boisot portrays the dynamics in I-space as continuous trajectories, either between identified points or regions or as generic trajectories through the space. Because Boisot uses I-space as a suggestive and descriptive framework, he doesn’t intend these trajectories to be interpreted as continuous transformations of the knowledge artifacts. He leaves open the issue about how to interpret trajectories at a micro level. 5) The Relevance of I-space to Innovation Research: As described so far, I-space can be seen as relevant for a broad range of social research (including, of course, sociology of baseball fans and players). But it may not be obvious how it serves the aims of innovation research because it has an obvious deficit, namely it doesn’t include the specific characteristics of any given set of knowledge such as you might find in a concept digram, semantic diagram, mind map, entityrelationship diagram, or any other representation of knowledge. Therefore it is silent regarding the role that any particular knowledge would or could play in any particular innovation process. It turns out that the I-space framework is highly relevant to innovation research because it allows us to analyze the general, abstract patterns of knowledge in innovation, regardless of the specific domain or relational context of that knowledge in the domain. Referring back to the pragmatic principles listed at the start of Section II-A, we can see a common pattern of trade-offs along each dimension, and also hints of interplay, feedback, and dialog between the extremes of each dimension. The most abstract generalizations suggest concrete implications (via deduction), while concrete experience can suggest new abstractions (via abduction) or be used to validate abstractions (via induction). Highly codified systems of information are made tangible through contextualized, tacit experiences, while tacit insights might suggest new or different ways of codifying information. Likewise, the collective knowledge of a community (e.g. culture, language, cohesion) shape the social interactions of individuals, who in turn engage with that collective more or less, while new collective knowledge and patterns can emerge from the interactions of individuals and small groups. Therefore, Boisot claims that I-space is relevant to innovation research because for innovation to be possible it is necessary for the community to exchange knowledge across a broad range
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in each dimension of I-space. In other words, Boisot claims that this far-ranging trajectory in I-space is a necessary condition for innovation. Boisot claims that processes that inhibit the flow of knowledge in I-space and thereby limit it to a narrow region will inevitably have limitations on the scope and nature of innovation that is possible. However, this far ranging trajectory is not a sufficient condition for innovation because of the specific characteristics of the knowledge and agents involved, as mentioned above. Furthermore, he makes a stronger claim that all innovation processes follow a similar trajectory through I-space, and he calls this the Social Learning Cycle (SLC). B. Social Learning Cycle
Fig. 2.
The Social Learning Cycle (SLC) in Information Space (I-space). (Source: Boisot [3])
Boisot’s Social Learning Cycle (SLC) [3] is a normative theory of knowledge creation, refinement, propagation, and diffusion. Boisot defines SLC as a cyclic trajectory (or set of trajectories) through I-space six identifiable and essential stages: 1) Scan the experience and insights of particular people to discover novel insights, solutions, or approaches. This knowledge will usually be tacit (i.e. hard to put into words) and specific to a given setting. 2) Codify that concrete/tacit knowledge so that it is easier to articulate and communicate 3) Abstract that knowledge in the form of general rules, laws, designs, or practices
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4) Diffuse that knowledge to community, still in codified and generalized form 5) Promote wide-spread absorption or adoption 6) deliver wide-spread beneficial impact The trajectory of SLC is meant to represent the flow of information as embodied in knowledge artifacts. It is not intended to represent flows or interactions in time, and thus some or all of the flows might be happening in parallel. Boisot does claim that the directional flow along the trajectory is an essential part of the theory. This paper is not intended as a critique or evaluation of the SLC, and thus I will not comment on its validity. A brief example will be useful to flesh out the SLC and show how it relates to innovation, using Figure 3 as a reference. This example will follow the general outline of Watt’s innovation in steam engine design described in Mokyr [1] but will omit many details for simplicity of exposition.
0.
ID problems in use
Deploy en mass
*+
Specify Improvement Opportunity
#.)
Design + Material + Production changes
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,)-./'&()*+
Mass produce
!"#$%&'()*+ Fig. 3.
Example innovation processes in the Social Learning Cycle (SLC) in Information Space (I-space). (Source: author)
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1) Identify (ID) problems in use – Watt identified evidence that the Newcomen2 steam engine had several problems in use, namely “awkward dimensions, a voracious appetite for fuel, and the difficulty early eighteenth-century mechanics had in achieving hermetic sealing.” ( [1] p 85) While many people were involved in building and using the Newcomen steam engine, presumably only a few people were paying close attention to the problems in use with an eye toward innovation. Furthermore, this learning used concrete and uncodified knowledge and information associated with the engines in use. Therefore, this process can be located in the lower quadrant of all three dimensions, as shown in Figure 3. 2) Specify improvement opportunity – Watt next made explicit the improvement opportunity by defining it in terms explicit codified terms: “. . . a desire to cut costs, minimize wear and tear, and extract‘the last drop of duty from the last puff of steam in his engine’.” ( [1] p 87) In terms of the I-space, the main difference between this knowledge and that of the first process is that it is more codified and explicit. Therefore this process is located higher on the Codification dimension, but still somewhat low in Abstraction and very low in Diffusion. 3) Design + material + production process changes – Once the improvement opportunity was codified, Watt designed his solution, which involved a “separate condenser from the piston cylinder, so that the later could be kept hot constantly. This separation greatly reduced the fuel requirements of the machine . . . ” ( [1] p 85) Watt also designed several other improvements, including steam-jacketing to keep the cylinder hot, a new transmission mechanism to convert reciprocating motion to rotative motion, and a parallel motion gear to enable a push-pull double acting expansion engine. These were primarily design changes and did not require new materials or substantially new production processes. In terms of the I-space and comparing it to the previous process, this design process is at a higher level of generality and abstraction and is therefore relatively high on the Abstraction dimension. It was also highly codified in the form of specifications and design drawings, and therefore it is high on the Codification dimension. Like the two previous processes, this one is low on the Diffusion dimension because the design was really only meaningful to Watt and 2
The Newcomen steam engine was the first commercially successful design for the first application, which was draining water
from mines.
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his immediate collaborators. 4) Mass produce – Once elements of the new design was complete, Watt next moved his steam engine into production, perhaps first as a prototype and then in quantity. In this process the knowledge artifacts of design are transformed into the engine as product in the marketplace. If Watt’s engine is to have any value or benefit to the broader community (i.e. mine owners and workers, and new potential customers and applications), it has to be accepted by customers in the marketplace. This acceptance is not assured even if it has technical or economic advantages over incumbents. In terms of the I-space, this process happens at a relatively high level of abstraction and codification, where the knowledge artifacts can include prices, contracts, terms and conditions, installation and usage procedures, etc. But it also happens at a fairly high level of Diffusion because marketplace acceptance is a collective phenomena. In terms of the I-space, this process is in the opposite corner than the first process and is high in all dimensions. 5) Deploy en mass – As the last stage in the SLC cycle for the Watt engine, customers need to transform the product that was purchased into a service that delivers value in use. In this process, the knowledge artifacts are associated with methods of operation and maintenance, etc. This knowledge is generally held in common among the community of Watt engine customers. In terms of the I-space, this process is high in Diffusion, but low in Abstraction because it is closely related to the concrete experience of operating the Watt engines in specific contexts. It is also relatively low in Codification because much of this learning is tacit and is passed on to others through learning-by-doing and direct observation. 6) (iteration) – The SLC continues to the first process as Watt and others learn scan for new improvement opportunities based often based on specific evidence and experiences. III. F ORMALIZING I- SPACE The approach for formalizing the I-space will be to specify ordinal functions for each dimension that preserve the definition of each dimension. Boisot provided informal definitions for three ordinal categories for each dimension – “High”, “Medium”, and “Low”. Our goal is to improve on these rough informal categories. The result will be three different (incommensurate) ordinal dimensions with fairly granular resolution, if not actually continuous in the sense of the real number line.
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With these dimensions, the formalized I-space will be an affine space rather than a Euclidean space. This will permit formal definitions characteristics and operations in I-space, including ’path’, ’trajectory’, ’similarity’, ’accessibility’, and ’proximity’ (an analog to distance). However, it will not allow distance, rotation, or rate-of-change operations, because of the ordinal and incommensurate nature of the dimensions. However this is not an obstacle to research because these operations are not necessary to study the SLC and similar theories. The second key element of formalization is to define ‘movement’ between point A and B within the I-space as one or more discrete ‘leaps’ rather than as a continuous movement through all intermediate points as is usual for Euclidean spaces. Thus, a path between two points in I-space is not a continuous transformation but instead is a continuous set of potential discrete transformations or leaps. A trajectory is a directed path, indicating transformation potential from A to B but not in reverse. Two paths are similar if they can be made isomorphic through translation operations in each dimension. If, in addition to translation operations, it is necessary to use scaling operations in one or more dimensions, then the two paths are only roughly similar. Two points in I-space are said to be accessible if and only if there is a trajectory between them. Proximity is more complicated to define, but basically means that, for all the paths involving three points A, B, and C, the paths between A and B can be said to be a subset of those between A and C, but not the reverse. Thus, point B is more proximate to A than C is to A. Using the operations of points, paths, accessibility, and proximity, it is also possible to precisely define regions in the I-space, which are roughly analogous to areas in Euclidean space. The remainder of this section will describe the ordinal functions for each of the three dimensions. A. Formalizing the Codification Dimension Boisot defines the Codification dimension by describing to three ordinal categories along with suggestions about definitions for the end points (extremes). Quoting from p 65 of Boisot [3]: •
High – Is the knowledge easily captured in figures and formulae? Does it lend itself to standardization and automation?
•
Medium – Is the knowledge describable in words and diagrams? Can it be readily understood by others from documents and written instructions alone?
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•
Low – Is the knowledge hard to articulate? Is it easier to show someone than to tell them about it?
In other sections, Boisot appeals to Shannon’s information theory to explain the Codification dimension. At the highest extreme of Codification, every message would be maximally compressed, and in the limit would consist of a single bit of information. This would only be possible if the coding scheme had a unique code for every possible knowledge state. At the other extreme of Codification, every message would be maximally uncompressed, and in the limit would consist of an infinite number of bits of information. These definitions of extremal points provide an important guidance on how to formalize the Codification dimension. In a metrological sense, these extremal points represent “unreachable infinities” in the sense that no physically realizable knowledge artifact could ever occupy these points. We will see similar characteristics on the other dimensions. Considering Boisot’s ordinal categories above, we can see three interrelated component phenomena that will need to be incorporated into the formalization. First, there will need to be a method for differentiating degrees of tacitness, where the knowledge is not explicitly represented in a coded message or information, or perhaps only in a suggestive form. This component will be labeled tacit, and it is generally applicable to the lower half of the Codification dimension. Second, there will need to be a method for differentiating degrees of coded expression in the usual sense of information theory as applied to messages, tokens, parameters, etc. This component will labeled codes and it generally applies to the center of the Codification dimension, though it has a role almost everywhere. Third, there will need to be a method for differentiating degrees of expressiveness within a language system that essentially encodes the rules or relationships between tokens. This will be labeled grammar and it generally is applicable to the upper half of the Codification dimension. I propose a mixture function to combine these three phenomena into an ordinal Codification dimension (see Figure 4). The mathematics of ordinal mixtures is somewhat complicated and therefore beyond the scope of this paper, but it can be formally defined using set theory and denotational semantics [4], [5]. Each of the three components is differentiated using ordered classes or mixtures. In the case of grammars, the Chomsky hierarchy [6] of formal grammars is defined according to the expressiveness of the language, where expressiveness is roughly the same as the sophistication or complexity of structures that are possible in the language. The
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Chomsky Hierarchy
interval ordinal
Cod e arti
nominal
cula
!"#$%&'(")*+$,-).$")*
ar
m Gram
ratio
High
Medium
te te t Taci inarticula
Low
Fig. 4. Schematic of the mixture function for Codification, showing the three components – Tacit, Code, and Grammar. (Source: author)
Chomsky hierarchy consists of the following (quoting from Wikipedia [6]): •
Type-0 grammars (unrestricted grammars) include all formal grammars. They generate exactly all languages that can be recognized by a Turing machine.
•
Type-1 grammars (context-sensitive grammars) generate the context-sensitive languages. The languages described by these grammars are exactly all languages that can be recognized by a linear bounded automaton (a nondeterministic Turing machine whose tape is bounded by a constant times the length of the input.)
•
Type-2 grammars (context-free grammars) generate the context-free languages. These languages are exactly all languages that can be recognized by a non-deterministic pushdown automaton.
•
Type-3 grammars (regular grammars) generate the regular languages. These languages are exactly all languages that can be decided by a finite state automaton. Additionally, this family of formal languages can be obtained by regular expressions. Regular languages are
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commonly used to define search patterns and the lexical structure of programming languages. Referring to the mixture schematic in Figure 4, Type-0 grammars would be at the top of the grammar component, followed by Type-1, Type-2, and Type-3. Type-3 grammars are sufficient for simple lists and category relationships, but Type-2 grammars are required for extensible hierarchies that are common in taxonomy structures. Contextual generative structures require Type-1 grammars, while Type-0 grammars are required for self-referential structures (i.e. selfmodifying, non-monotonic, etc.). At the upper extremal point in Codification, grammar dominates while codes (i.e. tokens) play an Infinitesimal role, but are still present. This is equivalent to saying that, at the extreme, nearly all the knowledge in the Codification is carried in the relational grammar system, which is maximally complex. Moving to the second component, codes, I differentiate it using the Theory of Scale Types first proposed by Stevens [7]. It, too, is a hierarchy in that variables in lower scale types can be expressed in higher types, but not the reverse. Moving from the top type to the bottom type (Wikipedia [8]): •
Ratio scale – A numeric scale using the real numbers where Zero has explicit meaning. Ratio scale variables permit all mathematical operations.
•
Interval scale – A numeric scale using the real numbers, integers, or natural numbers, but where Zero has arbitrary meaning. Interval scale variables permit addition, subtraction, equivalence (set operations), max, min, etc. but not multiplication or division.
•
Ordinal scale – A numeric scale using natural numbers where the number represents ordinal sequence. Ordinal scale variables permit min, max, and set operations, but not addition, subtraction, multiplication, or division.
•
Nominal scale – A non-numeric scale using names or labels for each value. Nominal scale variables only permit set operations.
Using the Theory of Scale Types allows us to formally define the degree of quantification in the key variables and parameters in any given codification scheme. Generally, only the Nominal scale applies when any degree of tacit coding is present, in the form of simple names or labels, and also category names. Moving to the third and final component of codification – tacit – this component can be differentiated as a mixture of two subcomponents (described in Dampney [5]): ‘articulate’ and
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‘inarticulate’. Informally, ‘articulate’ tacit knowledge are those elements that can be named or labeled for purposes of recognition and recall. Likewise, ‘inarticulate’ tacit knowledge is beyond naming or labeling and only resides in the perception and experience of a situated agent. Dampney [5] provides extensive lists of each of these subcomponents, and these lists might serve as a basis for an evaluation procedure. Since this is an ordinal mixture like Codification itself, similar mathematics would apply. At the lower extremal point in Codification, tacit dominates while codes (i.e. tokens) play an Infinitesimal role, but are still present. This is equivalent to saying that, at the extreme, nearly all the knowledge in absence of Codification is carried in the first-person experience, which is nearly ineffable. 1) Procedure: The following is a brief description of the procedure for evaluating the ordinal function for the Codification dimension. In the description that follows, the terms ‘mixture function’ and ‘submixture function’ refer to mathematical functions on ordinal mixtures that are suggested in the text above but have not yet been defined. This is future work. 1) For a given knowledge artifact, identify the primary or most important knowledge that it conveys or information that it provides. This will be defined relative to some usage context (epistemological domain) and community (social domain). 2) Identify the key tokens, variables, labels, or parameters. For each of these codes, identify the scale type that is used. 3) Identify the structural rules or grammar that are used to link tokens and variables together (e.g. order, hierarchy, web, etc.). For each of these grammars, identify the Chomsky type. 4) Identify tacit aspects or elements involved in the knowledge. For each of these tacit elements, identify both the ’articulate’ and ’inarticulate’ aspects. Use the submixture function to evaluate the tacit component. 5) Evaluate the Codification mixture function using the component evaluations – codes, grammar, and tacit. 2) On Continuity and Grain Size: The result of this procedure should be an ordinal value on the Codification dimension that is very fine grained (i.e. allowing many, many intermediate points between any to points) but not formally continuous as the real number line. However, the mixture function and components as it is described above, without additional elaboration, would not result in a fine grained ordinal dimension, and would certainly not be continuous.
19
However, if it is possible to define an ordinal function to operate within each component and level, then it would be possible to consider the ordinal scale as ‘fine grained’. For example, if two knowledge artifacts are being compared and they both use a Type-0 (regular grammar) to encode structural relations, then a separate ordering function needs to be applied to generate an order for all Type-0 grammars. This is certainly feasible, although there is no unique ordering function. In this example, Type-0 grammars can be ordered by the number of states needed in the reduced Finite State Automata (FSM) that accepts the grammar. Another ordering function would be the number of state transitions in the FSM. I conjecture that it is possible to define an ordering function for all of the components and subcomponents in the Codification dimension described above. If this is true and the ordering functions are feasible, then we can safely say that the Codification dimension is fine grained. In future work I will propose and evaluate ordinal functions for each of the components and subcomponents of Codification. However, we cannot say that it is a continuous dimension because we cannot prove that either countably infinite or uncountably infinite number of points lie between any two points. However, we don’t require continuity in dimensions because we will not be defining or using differential or integral functions. (See page 14). B. Formalizing the Abstraction Dimension Of the three dimensions in I-space, the Abstraction dimension is probably the hardest to formalize and specify in detail. The Codification dimension has information theory and computation theory as a basis, and Diffusion has social network theory as a basis. But what theory will provide the basis for Abstraction? In [2], Boisot suggests that semiotics is the best candidate and we shall follow his lead, though to different conclusions. Like the other dimensions, we will start by considering his suggested criteria for ordinal categories for Abstraction. Boisot defines the Abstraction dimension by describing to three ordinal categories along with suggestions about definitions for the end points (extremes). Quoting from p 65 of Boisot [3]: •
High – Is the knowledge generally applicable to all agents whatever the sector they operate in? Is it heavily science-based?
•
Medium – Is the knowledge applicable to agents within a few sectors only? Does it need to be adapted to the context in which it is applied?
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•
Low – Is the knowledge limited to a single sector and application within that sector? Does it need extensive adaptation to the context in which it is applied?
These guidelines all hinge on the definition of ‘applicable’ and ‘applicability’ of knowledge and the possible contexts for knowledge application. Unfortunately, possible definitions and interpretations of both ‘applicable’ and ‘context’ are too vague to serve as the basis for formalization. For example, if knowledge is ‘science-based’ or not may have as much to do with historical and cultural context as it does with the nature of the knowledge itself. His summary in [3] omits several important and nuanced concepts associated with abstraction that are described in [2]: ”Like coding, abstraction economizes on categories by an act of selection from competing alternatives. . . . Abstraction . . . can be thought of as a choice between competing hypotheses concerning which categories better capture a perceptual attribute. . . . The categories are not given a priori; they have to be discovered [or constructed] through a process of hypothesis creation and testing that make them emergent properties of the microstruture of cognitive processes. . . . Abstraction can thus never be reduced to a process of summarizing inputs; the operation has holistic properties . . . Beyond a certain level of abstraction, however, clusters of relationships sometimes acquire cohesiveness and a life of their own independently of perceptual attributes they play host to. They can form symbolic repetoire amenable to manipulation, and to further coding and economizing. Symbolic coding, however, is quite different business from perceptual coding. It allows one to build new structures out of elements that refer to other, more complex structures without requiring that these be represented in all their cumbersome detail. . . . “In sum, abstraction might be described as a move away from a process of manipulating images and other objects of experience that have been given form [in knowledge artifacts] – call this iconic coding – and toward operating with symbols that are more easily manipulated, stored, and retained in memory because they have been largely drained of perceptual content. The move from one to the other can be scaled. Semiotics, for example, gives us three useful points on such an abstraction scale.” ( [2] p 59-60) [glosses added] Boisot uses the schematic in Figure 5 to show how semiotic concepts of icon, sign, and symbol
21
can be used to subdivide and possibly scale the Abstraction dimension. I believe this is a very useful direction but that Boisot is mistaken in his application of semiotics. Before diving into semiotics, I will summarize the key points from this quotation. The icon
The sign
CONCRETE Fig. 5.
The symbol ABSTRACT
Boisot’s division of the Abstraction dimension using three concepts from semiotics. I do not agree with the specifics
here and offer an alternative scale in Figure .(Source: Boisot [2] p 60)
We may draw three important additional definitional elements from this extended quotation. First, that abstraction is tied to reasoning processes that link sense data (a.k.a. evidence) to higher level concepts such as categories and hypotheses. Thus our formalization of the Abstraction dimension should be explicitly tied to those reasoning process in both directions – from the specific to the general and from the general to the specific. Second, that abstraction is a holistic process such that the generalizations have an order and sense of among themselves – they reside in an intelligible system of categories, relations, and processes. Third, that the systemic dynamics of generalizations and abstractions give them ‘a life of their own’, that is abstract systems are often generative of new structures, new insights and new perspectives. They aren’t just static summaries of concrete perceptual processes. Boisot provides three references to support his ideas for applying semiotics to abstraction (Boisot [2], note 85, chapter 2): Morris [9], Peirce [10], and Eco [11]. Given the diversity of thought between these authors and different terminologies, I am not surprised that Boisot extracted only three general semiotic terms and definitions. Had he gone deeper, he would have encountered major difficulty reconciling Peirce and Eco, and even making sense of Peirce’s trichotomies and typologies as they might apply to a scale of abstraction. In the proposal that follows, I will be drawing almost exclusively from the semiotic theories and framework of C.S. Peirce, as elucidated by Frederick Stjernfelt in his recent book Diagram-
22
matology [12]. The reason for choosing Peircean semiotics3 is that it seems best suited for the purpose of formalizing a dimension of Abstraction to understand phenomena of social learning and innovation. Following Boisot, I believe the key to applying semiotics to a scale of abstraction is to view all degrees and forms of abstraction as semiotic – i.e. as sign systems – from the (almost) most concrete to the most abstract. I say “(almost) most concrete” because, strictly speaking, the extremal point on the low end of Abstraction will be the concrete experience without any signification, or with a vanishingly small degree of signification. But Boisot is mistaken when he applies the term ‘icon’ to concrete level of abstraction, the term ‘sign’ to mid-levels of abstraction, and ‘symbol’ to high level of abstraction. First, in Peirce’s terminology, both ‘icon’ and ‘symbol’ are types of signs, or more properly, sub-properties of some sign complexes. Peirce does distinguish between 1) an icon– a sign that “reflects qualitative features of the object” it is signifying; 2) an index – a sign that “utilizes some existential or physical connection between it and its object”, including direct pointing ; and 2) a‘symbol’ – a sign that “utilizes some convention, habit, or social rule or law that connects it with its object” [13]. Thus, to correct Boisot, it would be more in line with Peirce’s terminology to associate ‘index’ with ‘concrete’, ‘icon’ with an intermediate level of abstraction, and ‘symbol’ with a high level of abstraction. However, this simple application of Peirce’s terms has severe conceptual and practical problems and must be rejected. First, Peirce viewed signs as a arising from the interplay of three elements: 1) the object being signified, 2) the sign(s) themselves, and 3) the interpretant, the person or agent that makes meaning from the sign system. Furthermore, he viewed signs as almost always consisting of many elements which themselves might be classified as ‘index’, ‘icon’, or ‘symbol’. Indeed, over his career he developed ever-more complex typologies of signs based on various permutations and combinations of these trichotomies. It would only perpetuate Boisot’s error if we tried to map these typologies to the Abstraction dimension. To get more secure footing, we need to examine Peirce’s theory of reasoning and how it 3
Many scholars in semiotics and humanities criticize Peirce as being too complicated, too abstract, and too devoted to scholastic
distinctions between aspects of signs. If our concern were applying semiotics to literature, mass media, or popular culture, then these criticisms would certainly have merit. However, for our purposes Peirce’s scholastic approach to semiotics is an advantage, not a detriment.
23
relates to his semiotics. Most important, Peirce viewed the general nature of any given sign system as being tied to the ‘arguments’ it supported in the interpretant. Peirce’s use of the term ‘argument’ is not to imply a formal debate or dispute, but instead seems to mean ‘implications’, ‘inferences’, and ‘practical consequences’. This brings is back to Boisot’s informal guidance listed on page 19, above, which all used ‘applicability’ and ‘context’. With Peirce’s concepts added, then ‘applicability’ and ‘context’ become much more operational, I believe. Peirce viewed all reasoning as deeply semiotic, even inside the human brain at both conscious and unconscious levels. More important for our purposes, he viewed all formal abstractions and logic systems as being fundamentally semiotic. Adopting this approach will allow us to integrate the reasoning processes, the systemic nature of abstractions as sign systems, and the generative nature of abstractions, where operations on the sign systems can reveal latent knowledge that was not at first appreciated. If all abstractions are semiotic (sign systems), and all sign systems are intertwined with reasoning, then we need a way to distinguish reasoning that is more or less abstract, more or less concrete. Clearly, the form and structure of the sign alone will not suit us. Instead, we can use Peirce’s distinction between ‘corollarial reasoning’ and ‘theorematic reasoning’ ( [12] p 107-108). ‘Corrollarial reasoning’ refers to inferences or conclusions which may be read directly off the sign system with the proper general interpretation. A simple example would be a driving route from a street map, where the street map is the sign system and the driving route would be the ‘conclusions’ that are read directly off the map. In contrast, ‘theorematic reasoning’, requires the introduction of additional constructs not explicitly provided by the sign system. A simple example are the proofs in Euclid’s geometry, which often require construction of additional geometric objects beyond the initial object. More generally, ‘theorematic reasoning’ requires formulating and manipulating concepts and processes that go above and beyond the conceptual material evident in the sign system. In this sense, ‘theorematic reasoning’ is more inherently generative and fruitful than ‘corollarial reasoning’ in that it can ‘build castles in the air’ while ‘corollarial reasoning’ is limited to generating only permutations, combinations, and selections of existing conceptual material. Therefore, broadly speaking, we can say that the most concrete degree of Abstraction will involve only ‘corollarial reasoning’ while the most abstract degree of Abstraction will involve only ‘theorematic reasoning’, and in between these extremes will be a mixture, gradually increasing the proportion of ‘theorematic’ to ‘corollarial’ as abstraction
24
increases. To add some granularity to this mixture function, we can draw on a hierarchical category scheme devised by Peirce to characterize icon sign systems that functioned primarily through similarity to their objects, what he called ‘hypoicons’ ( [12] , p 90) with terms for additional subtypes of Diagrams originated by Stjernfelt [12] p 83, and also [14], [15]: 1) Image – a sign that represents its object through simple similarity or simple transformations 2) Diagram – a sign that represents its object through schematic relations, and those schematic relations are subject to continuous mental transformations and generative interpretations. ‘Diagram’ was Peirce’s umbrella notion for all schematic sign systems that supported reasoning, using a direct analogy to geometrical reasoning using diagrams of geometric objects. In Stjernfelt’s vivid language, diagrams are “moving pictures of thought” that gain their power through thought experiments using continuous transformations and other methods. a) Map – the most concrete type of diagram that yields conclusions through direct inspection. Examples include road maps, wiring diagrams, inventory lists, and dictionaries. b) Graph – a type of diagram that yields conclusions through a combination of direct inspection and some additional mental constructions. Examples include pie charts, bar charts, line charts, rule systems, ’how-to’ procedures and encyclopedias. c) Algebra – the most abstract type of diagram that yields conclusions only through additional mental constructions, and little or nothing by way of direct inspection. Nearly all formal symbolic systems of logic, mathematics, and computation fall into this category. 3) Metaphor – a sign that represents its object “through skeleton-like sketch of relations”, and thus only serves as a loose basis for interpretation or conclusions. Examples include common metaphors, similes and analogies, but also mythologies, cosmologies, and philosophical value systems. What is clear from these descriptions is that the top-most region of the Abstraction dimension will be ‘Metaphor’, while the near the low end will be ‘Image’, with a large region in the center for ‘Diagram’. The sub-types of Diagram can help us define subregions moving from concrete
25
toward abstract: ‘Map’, ‘Graph’, and ‘Algebra’. Bringing all these elements together, Figure 6 shows a schematic of the proposed ordering function for the Abstraction dimension. Sign System Type
Metaphor Algebra Reasoning Type
Th eo re
Ab st High
m
at
Di
ag
Graph ra m
Co
Map
ro lla Re ria ra as lR on ct ea io in so g n ni D ng Medium im
ic
Image Index
en si on Low
Reflexive Reasoning
Fig. 6. Schematic of the Abstraction dimension, showing two levels of analysis (sign system type and reasoning type) (Source: author)
Like the Codification dimension, we would like a fine-grained ordinal function for Abstraction. It is possible to subdivide the Index region, shown as the lowest region in Abstraction in Figure 6. For example, we could identify individual concrete perceptions as the least abstract point in the Index region, since they have almost no index sign associated. That is to say that people would have difficulty even pointing to them, let alone naming them, describing them or otherwise generalizing. This is so concrete and specific to a context as to be almost out of reach for any reasoning processes4 . At best, a person or agent could react to it through a subconscious conditioned response, which is also known as reflexive reasoning. 4
Indeed, the extremal point of least abstraction might be compared to the Zen Buddhist practice of “direct pointing” to pure
awareness which is beyond all concepts and percepts.
26
Above this extremal point, knowledge artifacts that are primarily index signs form the middle of the Index region. Examples would include an arrow as road sign, showing that a turn is coming soon, and also a scratch mark on the floor that could be a sign of a rough chair leg. Moving up in abstraction, pluralities and groups would be the next sign system that primarily functions as an index. Some types of categories might also function primarily as an index, though only those that are formed though conditioned association. Those categories that are formed by membership rules would better be described as Diagrams of the Graph type (i.e. because they are determined by rules plus member lists.) We could go on subdividing the Index region, but we are left with sub-subregions rather than an ordinal function, which doesn’t quite get us to the desired fine-grain resolution we are seeking. Considering now the Diagram region, which is covers the bulk of the Abstraction dimension, Stjernfelt ( [12] p 111) asserts that there is good reason to believe that there will not be a unique or ideal ordinal function for the Diagram region or it’s subregions due to its intrinsic ties to mathematics and logic: “As in any branch of research, the possible establishment of an inventory of rational subtypes will constitute a major progress. Unfortunately, no simple diagram taxonomy seems to be at hand, at least not referring to pure diagrams [i.e. the Algebra subtype] – for the very simple reason that the category of pure diagrams is coextensive with mathematics as such. This implies that the question of pure diagram taxonomies is inevitably tangled in the large questions of the foundations of mathematics. . . . The construction of a rational taxonomy of diagrams will be a major future challenge for (not only) Peircean semiotics.” ( [12] p 111) Therefore, I will offer an approximation procedure based on heuristics to estimate a mixture function of reasoning types. This will result in a fine-grained ordinal function for the entire Abstraction dimension, at the price of some imprecision and errors in the estimates. Switching to the reasoning types associated with the Abstraction dimension, the schematic in Figure 6 indicates that Abstraction is a mixture of three reasoning types: 1) Reflexive reasoning, 2) Corollarial reasoning, and 3) Theorematic reasoning. The last to are Peirce’s terms, while the first comes from behavioral psychology and cognitive science. The basic strategy to construct an ordinal function for Abstraction is to measure the ratio of the types of reasoning associated with the knowledge artifact in question, and then locate that proportion on the scale. But since
27
we will not be able to enumerate all of the reasoning systems that accept a given knowledge artifact (c.f. quote by Stjernfelt, above), we will need heuristics that will assist us in estimating the proportions. For this, we draw on research associated with human operation of machines and knowledge engineering. Both of these are practical applications of social learning. Rasmussen [16] takes a semiotic approach to human performance models to distinguish between processes that use abstract reasoning processes – those associated with “knowledge” – from those that use concrete cognitive processes – those associated with “skills” and “rules”. Using different terms than Peirce, Rasmussen describes the relationship between these three categories of reasoning and the associated semiotics: “At the skill-based level the perceptual motor system acts as a multivariable continuous control system . . . For this control the sensed information is perceived as timespace signals [‘Indexes’ or ‘Images’ in Peirce’s terminology] . . . These signals have no ‘meaning’ or significance except as direct physical time-space data. . . . Signals are [simply] sensory data . . . “At the rule-based level, the information is typically perceived as signs [‘Diagrams’ of the ‘Map’ or ‘Graph’ subtype in Peirce’s terminology]. The information perceived is defined as a sign when it serves to activate or modify predetermined actions or manipulations. Signs refer to situations or proper behavior by convention or prior experience; they do not refer to concepts or represent functional properties of the environment. . . . Signs indicate a state in the environment with reference to certain conventions for acts. . . . “[At the knowledge level,] to be useful for causal functional reasoning in predicting or explaining unfamiliar behavior of the environment, information must be perceived as symbols [In Peirce’s terminology: ‘Diagrams’ of the ‘Graph’ or ‘Algebra’ subtypes, or possibly ‘Metaphors’]. While signs refer to percepts and rules for action, symbols refer to concepts tied to functional properties and can be used for reasoning and computation by means of a suitable representation of such properties. . . . Symbols represent other information, variables, relations, and properties and can be formally processed.” ( [16] p 5-7) Rassmussen presents a hierarchy of abstraction based on these levels to help explain human
28
Functional purpose • Production flow models • System objectives
Purpose basis
Symbols
Abstract function • Causal structure • Mass/energy/information flow • Topology
``Generalized’’ functions • ``Standard’’ functions & processes • Control loops • Heat transfer
Signals
• Electrical, mechanical, chemical processes of components and equipment
Physical form • Physical appearance and anatomy • Material & form, locations, etc.
Physical basis
Physical functions
• Capabilities, resources • Causes of malfunction
Signs
• Reasons for proper function • Requirements
Rassmussen’s Levels of Abstraction in Human Performance with Machines
Fig. 7. Levels of Abstraction related to human operation of machines. Notice the similarities and parallel structure to the proposed Abstraction dimension in Figure 6. For example, “Physical basis” corresponds to the combination of Reflexive reasoning and Corollarial reasoning, while “Purpose basis” corresponds to Theorematic reasoning. (Source: Rasmussen [16])
performance with machines, shown here in Figure 7. Though the he uses different semiotic terms and the focus is on ‘causes’ and ‘reasons’ associated with machine operation, there are very definite parallels to the proposed Abstraction dimension in Figure 6, above. To the extent that a knowledge artifact is related to machine or tool design, use, or repair, we can draw specific guidance from Rassmussen’s descriptions of levels of abstraction. Working from the perspective of knowledge engineering, Gaines [17] presented a six level hierarchy of knowledge acquisition and transfer (Figure 8). The value of Gaines’ hierarchy for our purposes is that it further defines the types of reasoning along the Abstraction dimension proposed in Figure 6. Both models include ‘Reflexive reasoning’ at the bottom of the hierarchy. Gaines’ ‘Rule-based’ and ‘Computational’ levels seem to correspond to Corollarial reasoning, but only if we limit ‘Computational’ to mean optimization of a model using given inputs as Gaines suggests. Finally, Gaines’ ‘Comparative’, ‘Abstract’, and ‘Transcendental’ levels seem to
29
correspond to Theorematic reasoning. Thus, by adding information from Gaines’ hierarchy we can further refine and specify the types of reasoning proposed in Figure 6.
Gaines’ Hierarchy of Knowledge Acquisition and Transfer Knowledge acquisition 䚉 Action
Transcendental
6
distinction process 䚉 distinction creation
5
abstract models 䚉 specific models
4
alternative models 䚉 model selection
Transfer knowledge
Systemic principles
Use general principles to derive specific laws
Basic laws
Use basic laws to derive specific models
Analogical models
Derive knowledge from similar situations
Rational explanation
Read books, interview experts
Reinforcement
Work under expert supervision
Mimicry
Watch experts at task
Abstract
Comparative
Computational
3 2
Cultural acquisition
models 䚉 optimal action
Rule-based experience 䚉 action rules
Reflexive
1
construct 䚉 act
events
World
actions
Fig. 8. Hierarchy of Knowledge Acquisition and Transfer by Gaines, which for each level incorporates processes of knowledge acquisition, action, cultural acquisition, and knowledge transfer.(Source: Gaines [17])
1) Procedure: The following is a brief description of the procedure for evaluating the ordinal function for the Abstraction dimension. Unlike the other two dimensions, this procedure will primarily be based on heuristics. 1) For a given knowledge artifact, identify the primary or most important knowledge that it conveys or information that it provides. This will be defined relative to some usage context (epistemological domain) and community (social domain). 2) Use one or more of the following heuristic methods to estimate the ratio of reasoning types that are associated with or enabled by the knowledge artifact: a) Sign type method:
30
i) Enumerate the sign types associated with the knowledge artifact using the definitions described on page 24 above, and also [12] p 83, and [14], [15] as references. ii) For each of the enumerated signs, assign a weight according to its importance in the knowledge artifact. iii) For each of the enumerated signs, estimate the proportion of reasoning types that are compatible with that sign (c.f. Figure 6). iv) Multiplying each proportion by the weight of that sign, compute the weighted average proportions by reasoning type. b) Levels of Abstraction method: i) Enumerate the information conveyed by the knowledge artifact in terms of Rassmussen’s levels of abstraction hierarchy: Physical Form, Physical Functions, Generalized Functions, Abstract Functions, and Functional Purposes (c.f. Figure 7 and Rasmussen [16]). ii) For each information item, assign a weight according to its importance in the knowledge artifact. iii) For each information item, estimate the proportion of information that has a ‘physical basis’ (i.e. concerns causal processes) compared to ‘purpose basis’ (i.e. concerns reasons and purposes). iv) Multiplying each proportion by the weight of that information item, compute the weighted average proportions by reasoning type. c) Knowledge Acquisition and Transfer Hierarchy method: i) Enumerate the knowledge acquired or transferred through the knowledge artifact in terms of Gaines’ hierarchy levels: Reflexive, Rule-based, Computational, Comparative, Abstract, or Transcendental. (c.f. Figure 8 and Gaines [17]) ii) For each knowledge element, assign a weight according to its importance in the knowledge artifact or its use. iii) For each knowledge element estimate the proportion of reasoning types that are compatible with that knowledge element (c.f. Figure 6). iv) Multiplying each proportion by the weight of that knowledge element, compute the weighted average proportions by reasoning type.
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d) Frame Analysis method: i) Perform a frame analysis on the social actors who produce or consume the knowledge artifact (for description of frame analysis methods, see [18]). ii) For each element in each frame where the knowledge artifact is relevant, assign a weight according to its relative importance in the frame. iii) For each element in the frame, estimate the proportion of reasoning types that utilized in that frame element regarding knowledge artifact (c.f. Figure 6). iv) Multiplying each proportion by the weight of that frame element, compute the weighted average proportions by reasoning type. 3) With an estimate of proportion of reasoning types from one or more of the methods, above, combine them to reflect the ‘weight of the evidence’ to arrive at an overall estimate of proportion of reasoning types. 4) Use the following definitions and formulae to calculate the ratio-scale ordering function using the ratio scale score a: • •
a : Abstraction score in the range [0, 1] a0 : Abstraction score marking the end of the Image range (subtype of Diagram). This is somewhat arbitrary and will need to be chosen based on evidence or reasoning outside of the methods listed above. Setting a0 = 0.1 is probably a good starting place, so that the 0.9 of the Abstraction score is reserved for the Diagram and Metaphor sign systems.
•
r: proportion of reason type for Reflexive reasoning. If a = 0 then r = 1. If 0 < a ≤ a0 then r = 1 −
•
a . a0
If a ≥ a0 then r = 0.
c: proportion of reason type for Corollarial reasoning. If a = 0 then c = 0. If 0 < a ≤ a0 then c = 1 − r. If If a > a0 then c = 1 − t.
•
t: proportion of reason type for Theorematic reasoning. If a ≤ a0 then t = 0. If a > a0 then t = 1 − a +
•
a 1−a0
+ a0 .
Constraint: r + c + t = 1 by definition of proportions
a) If a ≤ a0 : a = (1 − r)a0
(1)
32
b) If a > a0 : a = a0 + t(1 − a0 )
(2)
The result of this procedure will be a fine-grained ordinal measure of the knowledge artifact on the Abstraction dimension, identifying a single point. Even though it is estimated through a ratio scale score, we still need to consider it an ordinal measure because the score is just part of an estimation procedure. Complex knowledge artifacts might yield a cluster or range of points. C. Formalizing the Diffusion Dimension Boisot defines the Diffusion dimension by describing to three ordinal categories along with suggestions about definitions for the end points (extremes). Quoting from p 65 of Boisot [3]: •
High – Is the knowledge readily available to all agents [in the community] who want to make use of it?
•
Medium – Is the knowledge available to only a few agents [in the community] or a few sectors [of the community]?
•
Low – Is the knowledge available to only one or two agents within a single sector [of the community]?
It is worth noting that the term ‘available’ is used in all three definitions. ‘Available’ word is less exacting than words such as ‘possessed’, ’learned’, or even ‘accessed’. All three of those alternate terms imply that the knowledge is known in some operational sense. Boisot’s softer term ‘available’ suggests instead that the knowledge has the potential to be known, both because it is readily accessible and is in a form that the person, group, or community can apprehend. Finally, the term ‘available’ includes many possible purposes for knowledge – learning, processing, acting upon, transferring, or utilizing consciously or unconsciously in some practical act. In this definition, Boisot differs from most of the literature on innovation diffusion, where the term ‘diffused’ generally refers to knowledge that is possessed or learned by a person, group, or collective. While it is tempting to formalize the Diffusion dimension by some ratio scale measure relative to the total population of the community, I believe that path is misguided. While the extreme values in Diffusion are a single individual (or subset of an individual) on the ‘Low’ end and the entire community on the ‘High’ end, intermediate degrees of diffusion depend critically on
33
social structures that mediate between individuals and the entire community. Here is a general hierarchy that accommodates most social structures: •
Community – All Individuals, Groups, and Groups of groups that, together, function as a community in some resepcts
• • •
Groups of groups – Two or more Groups that function as a social unit in some respects Groups – Two or more Individuals that function as a social unit in some respects Individuals – Single people or agents
!"#$%$#&'()
/0+&12)
Groups of groups
*+,,&"$-.)
!"#$%"&'(!")*'%"&'( Low Fig. 9.
Medium
High
Schematic of the Diffusion dimension, showing the four ordinal regions. (Source: author)
This hierarchy suggests a relatively simple ordering function based on the class of social structure that has ready access to a knowledge artifact, as shown in Figure 9. However there are some subtleties to attend to. The first issue is to distinguish between knowledge that is diffused at a community level and knowledge that is held by all community members but remains personal to each as individuals. My point is that not all knowledge that is ‘diffused’ can be properly defined as ‘Community’. For example, consider the 9/11 attack and the population of Manhattan as the community. While there is much knowledge of the 9/11 attack that are widely shared and readily accessible by all, it is also true that nearly everyone in Manhattan on that day had their own rich, unique personal experience at the ‘Individual’ level. In spite of commonalities between these personal experiences and imprints, they remain in their essence personal, uncodified, and concrete. The lesson here is to focus close attention on what knowledge is signified by the knowledge artifact. A newspaper page from 9/11 may signify the news of that day (= ‘Community’ knowledge that is codified and concrete) or it may signify the personal experience of one person if it was the last paper read by a person who died, and was thus cherished by her family (= ‘Individual’ knowledge that is uncodified and concrete).
34
Like the other dimensions, we are seeking a fine-grained ordinal scale for Diffusion, so it is necessary to look for ordinal functions for each of the four hierarchy levels listed above. For simplicity, the discussion will start with Individual (undiffused) and finish with Community (diffused). For the class ‘Individual’, we need a theory of subdivision that corresponds to ‘subset of an Individual’ without imposing strong restrictions on what we mean by Individual regarding age, gender, intelligence, education, role, culture or place in history. One useful theoretical lens is the theory of situated intelligence, which posits that who we are, what we know, and what we are able to do depends on the situation we are in. Obvious examples of situations include work, play, intimate relations, and social interaction. If we, as researchers, can enumerate the cardinal number of situations or classes of situations that are operational for an individual, and the total number of situations is finite (though, it could be very large), then we can define an ratio-scale function to subdivide the Individual region on the Diffusion scale, as follows: i=
s N
(3)
where i is the ratio-scale measure within the Individual range, s is the number of situations where the individual has ready access to the knowledge artifact, with 1 < s < N . N is the total number of situations for a given individual, which is finite but can be arbitrarily large. With this ordinal function for Individual, the minimum extremal point for Diffusion becomes a single situation within a single individual, who might have a very large number of situations. In other words, the knowledge artifact is accessible by a single individual, but only barely if the circumstance is only one of many possible circumstances. That individual would first have to put themselves in that situation in order to have access to that knowledge. Moving up the hierarchy, ‘Group’ is the next level (see Figure 9). A group may be a family, a team, a congress, or any organized collection of individuals. While the size of the group or the percentage of a group would be an easy ordinal function but it may not correspond to ‘knowledge accessibility’, as discussed above. Here we encounter a difficulty in creating an ordinal function because groups often have internal structures, especially if they are large. For example, if the group has a simple hierarchy structure, then if the knowledge is accessible to the top members of the hierarchy, then it is in principle easily accessible to all members below them in the hierarchy, but the reverse is not true if it is an authoritarian hierarchy. Group structures, as topologies, are
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not easily ordered by a single function. The most condensed space of organization forms that I have found [19] consists of a three-dimensional simplex, shown in Figure 10. Clearly it is not possible to reduce this space to a single dimension without losing information. However,
Inn
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ns
rgy ity il tab un
co
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Ac
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Sy
ov ati on ist en cy
Cooperation
Global Perspective Local Sensitivity
Autonomy
Fig. 10. A simplex space of organization structures with three ratio-scale dimensions in the range (0,1): Autonomy, Authority, and Collaboration. Three example organization structures are given for the extremes in the space. In principle any organization structure can be identified with a single point or small region of points in this space. While this three-dimensional space is a very compact representation, it serves to demonstrate that it may not be feasible to measure organization structures on an even simpler space – a single dimension. (Source: author, based on Keidel [19])
because our focus here is on diffusion within a group, we do not need full information about the organization structure even at a general level. Instead, we can focus on the diffusion properties of organization structures. For this, we can use social network theory to define a measure that would be applicable to any group structure. In social networks, communication speed is often governed by the shortest path between any two nodes, where a path is measured by ‘hops’, i.e. the number of edges that need to be traversed between nodes A and B. Considering a social network as a whole, one possible measure would be the average shortest path length for all nodes to every other node. If we identify the individuals in a group who have immediate access to the knowledge signified by the knowledge artifact, we could compute the average minimum path from those individuals to all other individuals. In simple, highly regular structures, this can easily be calculated by inspection. More complex and quasi-random structures would require computer assistance. For a more sophisticated measure than accounts for parallel paths of communication,
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the measure of conductance could be used. This is a direct analogy from electrical circuit theory, and the calculations are of a similar complexity (see Wikipedia [20]). Both average minimum path and conductance could be normalized by the group size to produce a ratio-scale ordering function for the ‘Group’ range of the Diffusion dimension. This scale would have boundary values of 1.0 for minimum diffusion (when the average minimum path is equal to the number of members of the group) and 0.0 for maximum diffusion (zero minimum path length because there are no members of the group who do not have immediate access to the knowledge). Moving up to the Group of Groups region in the Diffusion dimension (see Figure 9), we can use the very same ordinal function as defined for Groups, except that this time we substitute groups for individual people or agents. The Groups of Groups region is important to understand social learning in meta-organizations such as value chains and value networks, industry groups, alliances, etc. The final region in the Diffusion dimension is ‘Community’ (see Figure 9). While it may be tempting to treat Community as a single point on the dimension, i.e. as a totality or unified whole, it is fruitful to differentiate ‘degrees of the whole’ that go beyond any Groups of Groups but still not constitute complete diffusion in the community. We can apply the same ordinal function as Groups and also Groups of Groups – the average minimum path between those members who have immediate access to the knowledge and those who do not. In many or most cases, the analyst will not know the exact social network that links a community. However, for measurement purposes, it is probably sufficient to estimate and even idealize the social network structure, then apply diffusion-of-innovation analysis applied to well-known structures (e.g. completely connected graphs, random graphs, ‘small world’ graphs, etc.). This procedure will allow a fine-grained ordinal function for the Community region that will be sufficient for most research purposes, I believe. 1) Procedure: The following is a brief description of the procedure for evaluating the ordinal function for the Diffusion dimension. 1) For a given knowledge artifact, identify the primary or most important knowledge that it conveys or information that it provides. This will be defined relative to some usage context (epistemological domain) and community (social domain). 2) Identify whether the knowledge signified by the knowledge artifact is accessible to an Individual, a Group, a Group of Groups, or to the Community as a whole.
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3) If ‘Individual’, then evaluate the ordinal function for individuals by estimating the proportion of situations where the individiual has ready access to the knowledge signified by the knowledge artifact. 4) Otherwise, if ‘Group’, ‘Group of Groups’, or ‘Community’, then evaluate the ordinal function as follows. First, estimate the social network and location of ‘immediate accessibility’ within the social network. Then calculate the average minimum path between those members who do not have immediate access to those members who do. Normalize by dividing by the group or community size. The result of this procedure will be a fine-grained ordinal measure of the knowledge artifact on the Diffusion dimension, identifying a single point. Complex knowledge artifacts might yield a cluster or range of points. Though the ordinal functions within each region (e.g. Individual, Group, etc.) are ratio scale, the overall scale for this dimension is ordinal. IV. C ONCLUSION This report focused on research method associated with social learning in innovation communities. The main goal was to formalize the Information Space (I-space) framework of Boisot to support empirical research and computer modeling. I believe that I have succeeded in achieving these goals, though the formal I-space still needs to be tested in research practice. A schematic of the full I-space is shown in Figure 11. These results were achieved by identifying an ordinal function for each dimension that was appropriate to that dimension, grounded in appropriate theory. The same general strategy was used on all three dimensions, namely to find a set of ordinal or hierarchical categories to subdivide the dimension, and then find an ordinal function appropriate to each category/region. This approach has the benefit of providing a fine-grained ordinal measure across the whole dimension without attempting to justify a continuous ratio scale measure. In one case, the Abstraction dimension, we did propose a ratio scale score, but only as an approximation method for the ordinal function. By defining a formal fine-grained ordinal measure on each dimension, it becomes possible to rigorously define and analyze points, paths, trajectories, regions and even gradients in I-space. This should enable hypothesis testing regarding the necessary conditions for innovation, especially important macroinventions that have a restructuring effect on industries and societies.
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at
ic
Graph ra m
io n
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Co ro lla Re ria as lR on ea in so g ni ng
ar
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m Gram
Codification Dimension
Chomsky Hierarchy
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Diffusion Dimension Individual
Groups
Groups of groups
Community
Reflexive Reasoning
Fig. 11.
A schematic of formalized I-space. (Source: author)
There are several limitations in the methods presented in this paper. First, I did not develop the mathematics of ordinal mixtures necessary for the Codification dimension. The mathematics of ordinal mixtures is somewhat complicated and therefore beyond the scope of this paper, but it can be formally defined using set theory and denotational semantics. This will be done in future work and is essential before the framework can become operational as a research tool. Finally, the formal framework has not yet been tested using case studies to validate and to refine the ordinal functions for each dimension. This testing will be necessary to convince other researchers that the framework is valid, and also to help explain how to use it in practice. When complete and validated, formal I-space framework presented in this report should be very helpful in empirical research to test hypotheses related to Boisot’s social learning cycle or social learning and collaborative innovation theories. It should also be very helpful in computer modeling such as Agent-based Modeling (ABM) where cognitive agents interact socially in the context of innovation processes. The formal I-space will provide a computational landscape for
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interaction similar to the fitness landscape used in evolutionary computation studies. ACKNOWLEDGMENT This research is supported by a grant from the National Science Foundation, grant no. SBE0915482. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. I wish to thank Professor David Schum for his guidance, enthusiastic support, and flexibility over two semesters of courses. RCT November 29, 2012
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