Boden's theory of creativity explains the creative process in terms of exploration in, and transformation of generatively-represented conceptual spaces [Boden, 1990, 1998, 2003]. An advantage of the theory is that its terms of reference are ordinary computational processes, readily understood in terms of conventional models. Her core definition however, which attempts to identify creativity specifically with transformation (generation of new spaces), has been questioned, since both transformation and exploration are ultimately applications of search. This paper attempts to follow the suggestion of one reviewer [Bundy, 1994] to re-cast the definition in quantitative terms, using concepts of computational complexity. The approach involves giving the complexity of Boden's search processes a formal evaluation and using this as the foundation for a revised definition.
Keywords: creativity, complexity, conceptual space, representation
Unlike Koestler's theory, where the envisaged mechanism was idiosyncratic, Boden's theory dealt in terms of conventional computational structures and processes. Moreover, her theory described mechanisms which could, in principle, be constructed using standard computational devices. This gave Boden's theory a clear advantage. As Boden notes `The thought processes [Koestler] described do happen, and they do seem to be involved in creativity. But because how they happen was not detailed, he did not fully explain how creativity is possible.' [ibid, p. 24]
Boden's starting point for the development of her account is the observation that the concept of creativity contains a paradox. By definition, creativity creates, i.e., it produces something new. But if we are committed to a mechanistic account of the world---no miracles allowed---we believe that everything that occurs is predictable in principle. We also believe that any new thing must be constructed from existing components. This implies that nothing can ever be intrinsically new. How, then, should we reconcile the definitional requirement that creativity produces novelty with the assumption that there is no such thing as novelty? Boden's aim is to deal scientifically with this question.
She takes it as given that creativity is always mediated by conceptual development of some form. Any creative act is thus always founded on conceptualisation or the realisation of a `point' within a particular `conceptual space'. But, she notes, there are two ways in which this might happen. If the conceptual space has an existing mental representation, realisation of a new point is simply a matter of identifying a new location within that space. If no such representation exists, then realisation of a new point necessarily involves construction of the representation as a preliminary step.
For Boden, this offers the means of distinguishing two forms of conceptualisation: a straightforward form, involving the identification of a new point in an existing space, and a more complex form which involves as a preliminary step the construction of the relevant conceptual space. The process of identifying a new point she terms exploration; the process of generating a new space she calls transformation. Her key idea is then to use the distinction between exploration and transformation to define what is to count as true creativity.
`We can now distinguish first-time novelty from radical originality' she writes. `A merely novel idea is one which can be described and/or produced by the same set of generative rules((For reasons that will be explained, it can be assumed that mental representation of a conceptual space must be generative in character.)) as are other, familiar ideas. A genuinely original, or creative, idea is one which cannot.' [Boden, 1990, p. 40] A new concept, then, is to be considered genuinely creative just in case its construction involved some element of transformation.
For Boden, this way of framing the definition effectively resolves the creativity paradox. A concept whose construction involves some element of transformation cannot be generated on the basis of existing mental representations. In this sense, it is mentally `impossible'. Boden then proposes we can justify saying a concept is `new' on the grounds that it was previously `impossible'. Or, as she puts it `To justify calling an idea creative ... one must identify the generative principles with respect to which it is impossible.' [ibid., p. 40]
With this transformation-based definition, Boden establishes an intuitively clear characterisation of the process of creativity. However, consideration of further evidence compels her to take steps to generalise the characterisation in several ways. First, she observes that ordinary exploration of existing conceptual spaces may also be creative. With regard to mathematics for example, she notes how the `creative mathematician explores a given generative system, or set of rules, to see what it can and cannot do.' [ibid., p. 45]. The recognition that exploration may be creative also surfaces in her comment that `creativity is a matter of using one's computational resources to explore, and sometimes to break out of, familiar conceptual spaces.' [ibid., p. 108] However, it is not Boden's intention to broaden the definition to include brute-force (uninformed) exploration. Exploratory processes are only to be considered creative if they are guided in some way by `heuristics' or `maps'.
Boden also makes a point of allowing that transformation may constitute creativity in its own right, regardless of any ensuing concept development. This is evidenced in her comments on creative mathematics: `By creative mathematics, I do not mean adding 837,921 to 736,017 to get 1,573,938 ... I mean producing new generative systems, new styles of doing mathematics.' [ibid. p. 45] The construction of new generative systems, then, would appear to fall into the category of creative processes, regardless of whether any concepts or ideas within that new space are ever actuated.
The indications are, then, that Boden wishes to generalise her original definition so as to allow that creativity may involve either
However, caution is in order. In neither edition of her book does Boden go so far as to explicitly differentiate a `weak' from a `strong' definition. In fact, in the first edition, she offers no final count of the number of different types of creativity she has identified. It seems to be her intention to distinguish the two types noted, and this is certainly a common interpretation. Yet in the `nutshell' summary of her theory, added as a prologue to the second edition [Boden, 2003], and in [Boden, 1998], she states that her account distinguishes three main forms of creativity, these being exploration, transformation and combination. Adding to the uncertainty is the observation that only the strong definition has the power to resolve the creativity paradox, arguably forcing us to recognise not two forms of creativity, or three, but one: transformation.((Boden's own remarks about the definition tend to heighten the element of doubt. The characterisation is `vague' she writes, and no more than `intuitive talk'. For her, it cannot stand on its own: `Anyone hoping for a scientific explanation of creativity must be able to discuss mental spaces, and their exploration more precisely.' [ibid. p. 73]))
There is a choice to be made, then, regarding the definition. There is the original formulation, which seeks to identify creativity specifically with transformation. Then there is the more general characterisation which allows that exploration may figure. Finally, there is the view from the post-1998 publications which adds `combination' as a possibility.
The original formulation is clearly stated, but as Boden herself has shown, not completely general. Her broadened characterisation on the other hand may be more general, but it is too all-encompassing to be useful as a definition. As for the inclusion of `combination' as a separate type of creative process (as suggested in the second edition) this would seem to be a matter of explanatory expedience. As Richie notes, the distinction between combinational and exploratory is `hard to pin down' [Richie, 2001]. Indeed, any combinational process must presumably operate through exploration of a space of possible combinations. So it seems reasonable to view combinational processes as subsumed within the exploratory category.
Insofar as there is a definition underpinning the theory, then, it has to be the `strong definition'. But this raises a difficulty. Even taking on board the withdrawal of any generality claim, the definition cannot be regarded as a firm foundation for the theory. It contains an ambiguity which undermines its application in practice. Although the two processes Boden describes appear to be distinct and separate, in computational terms they are not. They are both applications of search. [Wiggins, 2001]
Boden herself notes [Boden, 1995, p. 161] that many of the reviewers of her account have pointed to problems with the definition, and specifically to the problem of ambiguity. Samples are to be found in Fetzer [Fetzer, 1994] O'Rourke [O'Rourke, 1994], Bundy [Bundy, 1994], Ram et al [Ram et al. 1994] and [Ram et al. 1995] Treisman [1994] and Weisberg [1994]. Several of these writers have noted the fact that transformation is really just a type of search, so would seem to logically belong in the same category of process as exploration [e.g. Ram et al. 1995, p. 114]. Wiggins has used a rationalisation of Boden's theory to demonstrate the same point more formally [Wiggins, 2001]. Some have noted that transformation is a form of `meta-search'. Schank and Forster describe application of the definition as a `mare's nest' [Schank and Forster, 1995, p. 138] while Turner [Turner, 1995, p. 152] notes that while Boden wants to view computers as potentially creative, by her `definition of creativity, it is impossible for a computer program to be creative.' [Turner, 1995, p. 153]
In her responses to this criticism, Boden pleads guilty to the charge that the definition is insufficiently clear. `Most of the reviewers point out, quite rightly,' she says, `that my definition of creativity ... was vague.' [Boden, 1994b, p. 559] Her defence is that the necessities of writing for a general audience dictated the limiting of technical content. But, to be fair to the reviewers, the problem they highlight is not so much that the definition is vague, but rather that it no computational foundation.
According to the proposed rule, any thought which is classified as creative on the grounds that its construction has involved the creation of a new conceptual space, may also be regarded as uncreative, on the grounds that the creation of that space itself involved ordinary search (or meta-search as we might call it) in another, over-arching conceptual space. Thus, classification done using Boden's definition will generally produce two, contradictory results: one positive, the other negative.
In later writings, Boden concedes there is a problem but suggests that it results from mis-application of the criterion. `If we are to apply [my definition] to the entire resources of the person's mind (or the computer's program), of course we get an inconsistency' she says. [Boden, 1995, p. 163] But the definition's inconsistent results do not come about as a result of mis-application. They result from the fact that transformation, as a process, cannot easily be distinguished computationally from exploration, since both are forms of search. Search is just the generation of candidates in a space of possibilities and as Bundy has noted `Transformation is [simply] a kind of generation.' [Bundy, 1994, p. 534]
There is a deep problem, then, with the definition. But it would be a mistake to let these difficulties undermine the theory, which has proved its value in many ways. The essence of Boden's definition is that creativity involves some sort of special, representational effort. And there are other ways in which we might carve out what this `extra effort' might involve. Wiggins has proposed the idea that it might take the form of meta-traversal of conceptual space representations [Wiggins, 2001], arguing that this might form the basis for a different interpretation of the meaning of `transformation'. An alternative approach involves re-working the definition in terms of some notion of complexity, i.e., a computational measure of `effort'. This line of attack was suggested by at least one of the initial reviewers [Bundy, 1994] but, to my knowledge, has never been followed up.
Doing so is the specific aim of the present paper. The plan will be as follows. The next section will review the core concepts in Boden's theory and clarify their technical aspects. Following this, there will be an analysis of the complexity of conceptual-space search. Finally, the complexity analysis will be used for purposes of developing a quantitative revision of Boden's creativity definition.
A conceptual space in her terms is simply a set of associated concepts, assumed to be represented generatively. In computational terms, the conceptual space is a virtual set of objects representing concepts. But the generative nature of the representation is essential. Any form of explicit, `point-by-point' representation for conceptual spaces would be impractical and, in the case of an infinite space, impossible. The generative approach overcomes these difficulties. As Boden comments `a generative system [has] ... the potential (in principle) to generate every location within the conceptual space. The number of these locations may be very large, even infinite.' [ibid. p. 78]
In operation, any generative system follows a particular methodology. To a set of originating objects---the `prototypes' or `primitives'---it applies processes which have the effect of combining those objects to form new objects. If there is the potential for recursion, these objects may then become constituents in the construction of further objects and so on. Any generative system must therefore embody (a) a set of originating objects and (b) functionality for the construction of new objects from existing objects.
Programming languages provide a convenient framework for the implementation of such systems. For example, consider this sequence of instructions:
To each number between 0 and 9 apply the next 2 instructions To each number between 0 and 9 apply the next instruction Print the first number followed by the second numberThis program has the effect, when `executed', of printing out all 100 points in a 10x10 2-dimensional space; i.e.,
0 0, 0 1, 0 2, 0 3, 0 4, 0 5, 0 6, 0 7, 0 8, 0 9 1 0, 1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 7, 1 8, 1 9 2 0, 2 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2 7, 2 8, 2 9 . . <60 pairs omitted> . 9 0, 9 1, 9 2, 9 3, 9 4, 9 5, 9 6, 9 7, 9 8, 9 9The program effectively provides a generative representation for the 2-dimensional space based on the integer range 0..9. The primitive objects are the integers. The construction method is the procedure for printing pairs of digits.
Boden's preferred example of a generative representation is the grammar. And indeed, grammar notation is a particularly convenient way of detailing generative systems. In writing out a grammar we only have to specify the bits of information which are essential for object construction---*what* object results from what combination. Everything else is implicit.
Consider this:
S -> NP V
NP -> Det N
Det -> the | a | one
N -> man | woman
V -> ran | walked | sang | laughed
The first two lines of this grammar are rewrite rules showing
how the object named on the left of the arrow can be
constructed by combining objects named on the right. The
remaining three lines are lexical rules which show how each of
the words on the right is an example of the object named on the
left. We may think of the whole as a generative representation
for all the sequences of words which result from all the
different ways of constructing an `S' object. These include
`the man ran', `the man walked', `the man sang', `the man
laughed', `the woman ran', `the woman walked' and so on.For present purposes, we are interested in generative representation of conceptual spaces rather than languages. These have special features which go beyond what we expect to find in a grammar or program representation. As noted, any generative system must embody a set of originating objects and functionality for the formation of new objects from combinations of existing objects. In the case of the conceptual space the constructed objects must obviously be concepts. But what of the originating objects? Ultimately, these are simply `phenomena'; but the system's representation of any phenomenon can only be itself a concept. Originating and constructed objects are thus all of the same type---they are all just concepts.
Possible forms of conceptualisation.
If there is more than one way to assemble any given set of components (i.e., more than one way of placing them in a relationship) then the compositional process has several forms---one for each relationship. If only simple conjunctions can be formed, the compositional process can only be applied in one way and always has the same result.
figure-shown shows the various ways concepts may be formed. The
filled circles at the bottom of the figure represent the primitive
or originating concepts. Empty circles in the layer above represent
categorical concepts. Circles containing internal structure
represent compositional concepts. In this scenario, the
conceptualiser is assumed to be capable of forming two distinct
forms of compositional concept, one based on the relationship
and the other based on the relationship 
.
The organisation of the intersections of the connecting lines in
these concepts is intended to represent the way in which the
concept imposes a particular relationship over the components. In
the case of the categorical concepts the arcs combine, representing
the fact that the components in this case are treated as members of
a class of alternatives. In the case of compositional concepts, the
arcs are independent, representing the fact that the components are
constituents.
Possible avenues of concept development.
More generally, we have the deep conceptual space. This results when concept-construction may be recursive. The result of any compositional process is a new concept which may be used in a further round of concept construction. Thus, with compositional construction, there is always the potential for incremental, hierarchical growth of the space. The effect is illustrated in figure-shown.
Possible avenues of recursive concept development.
The distinction between deep and shallow conceptual spaces is qualitative rather than quantitative. As a rule, the deep conceptual space contains more concepts than a shallow one. Often it will be infinitely large. But there may be constraints which overturn the normal balance. Treating the sample grammar from above as a concept-generation system, one would have to say that it is `deep' on the grounds that it makes use of recursive compositional rules. However, constraints are in place which limit the recursion. The total number of constructs is therefore finite. Were one to introduce a new rule so as to allow for infinite recursion, however (S -> N for example), the situation would be altered.
The programmatic representation of 2-dimensional space, in contrast, is in the `shallow' category due to the fact that it makes no use of recursive construction. Compositional construction is featured but only a tiny fraction of the possible combinations of primitives feature may appear in any generated construct---in particular the ones containing exactly two objects. The total number of these is related to the size of the represented space. But this may be arbitrarily large. Thus, this type of 'shallow' space has the potential to be larger than the `deep' space defined by the grammar.
Assume we start with n originating concepts and make use of unconstrained construction methods, i.e., composition/categorisation applied to any combination of components. The undeveloped conceptual space, in which construction plays no role, contains just the originating n primitives. There are no derived concepts.
Introducing a categorical construction method produces a space with a distinct concept for every subset of the primitives. The maximum number of concepts is then
Let us say both categorical and compositional construction methods
are available. At the bottom level of the hierarchy we have the
n primitives. At the layer above, the number of concepts is
some multiple of , since the number of possible
combinations of n objects is , and our assumed
system can generate a fixed number of concepts for each
combination. Let's say the system is capable of generating (for
each combination) a categorical concept and r compositional
concepts. In other words, it is capable of arranging the components
as a class or in r different relationships. In this case the
total number of concepts which can be generated
is multiplied by 
. Given we are
exponentiating to base 2,
this can be expressed as
With r methods of compositional construction, exactly
of any new set of concepts will be class-based.
Therefore in specifying the true number of potential concepts at
the next level we should subtract
possible
combinations are singleton sets, i.e., they simply yield
`stand-ins' for nodes at the layer below. (We could discount
these by subtracting 
.) However, if we wish to allow
that construction may utilise concepts from different levels of
the hierarchy---and at present we have no reason not to---it
simplifies matters if we allow the singleton sets to remain.
This way, every level of the tree contains a `stand-in' for
every node at every level below. The possibility of cross-level
combination is automatically taken into account.
To render the formula in a general form is now straightforward.
If 
is set equal to the number of originating
concepts, the number of potential concepts 
at the
i'th level of the hierarchy may be calculated using the
recursive formula
As may be clear, the unconstrained, deep conceptual space grows at a prodigious rate, driven by two distinct exponential forces.((In addition to the explicit exponential value, there is a second multiplicative effect implemented through the recursion.)) After construction has proceeded above the initial few levels, the number of potential concepts becomes astronomically large. In any real context, only a small minority will have any value. But all are syntactically feasible.
We can illustrate the strength of the effect by considering an
example. Were we to start out with just two primitive concepts and to
apply just one method of (unconstrained) composition in addition to
(unconstrained) categorisation, we could develop four concepts at the
first level of construction and 16 concepts at the second level. At the
third level, there are more than 65000 to take into consideration but at
the fourth level there are nearly than 
. Processing
these at even the implausible rate of one quadrillion per second, it
would still take far longer than the estimated age of the universe to
get through them all. It can be inferred that any sort of brute-force
exploration of conceptualisations is impractical.
This prompts the making of a distinction between conceptual spaces and conceptual universes. An agent's conceptual universe may be defined as the set of concepts that may be developed from the agent's original concepts and construction methods, i.e., whatever concepts it possesses at the start of its concept-using existence.((Wiggins makes a similar distinction in setting up his formalisation [Wiggins, 2001].))
This conceptual universe contains every possible concept that the agent might develop. It also subsumes every possible conceptual space that the agent might develop. In other words, the sub-spaces of the conceptual universe are the agent's potential conceptual spaces. A schematic illustration of the situation appears in figure-below.
Conceptual spaces as sub-spaces within a conceptual universe.
Boden is committed to the idea that the two processes are distinct. Indeed she goes to some lengths to show how existing artificial intelligence models allocate functionality between the two. The examples she provides cover a wide territory. Working forwards from her Chapter 6 [ibid, pp. 112-216], we see that creative, conceptual-space operations may involve `pattern-completion' and `contextual memory' as featured in connectionist networks [Rumelhart et al. 1986]; `reasoning', and `problem solving' as featured in traditional problem-solving systems; combining of `general and specific knowledge', `constraint satisfaction' and the exercise of `judgement' as featured in Harold Cohen's AARON system [Cohen, 1981].
It may involve operating in terms of complex, nested, hierarchical formulations, as in Johnson-Laird's jazz improvisation program [Johnson-Laird, 1988]; the application of `knowledge of planning' as in TALE-SPIN [Schank and Riesbeck, 1981, pp. 197-258]; `analogical thinking' on the basis of `multiple constraint satisfaction' and a recognition of `semantic similarity' as featured in ACME [Holyoak and Thagard, 1989]; the creation and mapping of analogies (using generate and test) as featured in ARCS [Holyoak et al. 1988]; induction and learning, as featured in systems like AQ-11 [Michalski and Chilausky, 1980] and ID3 [Michie and Johnston, 1984, pp. 110-12]; searching for new rules, as featured in meta-DENDRAL [Buchanan et al. 1976], or for quantitative relationships, as featured in `discovery' systems such as BACON [Langley et al. 1987] or for qualitative laws, as featured in GLAUBER [ibid]; defining second-level theoretical terms as in later versions of BACON [ibid]; `hypothesis generation', `heuristic application' and `concept transformation' as featured in AM [Lenat, 1977]. Last but not least it may involve rule-specialisation, as featured in EURISKO [Lenat, 1983].
This is certainly a broad catalogue. But the published descriptions for these systems make little or no use of Boden's `conceptual space' terminology as such. Indeed, in most cases, no reference is made to operations on any kind of concept packaging structure. Where these systems make use of concept representation, they do so on the basis of conceptual content rather than on the basis of conceptual-space membership. In general conceptual spaces are not significant as an implementational construct in these references.
There would appear to be nothing in the example set, then, which requires maintenance of the distinction between exploration and transformation. In fact, consideration of the likely representational requirements suggests, rather, that enforcing the distinction in a conceptual agent might impose prohibitive resource costs. The originating concepts for any new conceptual space must clearly be represented somewhere in the overall conceptual system. But they cannot be located within the conceptual space itself, for fear of infinite regress. The content of a primitive of a new conceptual space is thus necessarily located outside the space itself. Since every concept within a conceptual space ultimately references the content of its primitives, content is then always to some degree cross-referenced against other conceptual spaces.
Exacerbating the effect is the fact that the situation is `recursive'. Concept A may invoke concept B, which invokes concept C, and so on. We therefore have to expect the content of a given concept to be distributed across a series of spaces. In an extreme case, it might be `smeared' across the entire conceptual system.
Achieving independent representation of conceptual spaces would necessitate some sort of `de-smearing' operation, i.e., introduction of a mechanism which has the effect of encapsulating within a given space all the chains of reference underpinning the content of member concepts. But except in the most trivial scenarios, the combinatoric properties of any such operation would generate exponentially increasing resource requirements. We have to assume, then, that conceptual spaces cannot be represented ultimately as independent objects. Rather, they must be acquired from a representation in which constituent structure serves to represent content in the usual way. At its most basic level, then, it would seem that a concept-representing system cannot be structured as a set of independent conceptual spaces. It must be structured in the familiar way, as a hierarchy where between-concept connections serve to reference content.
Two arguments thus converge in support of the idea that the distinction between exploration and transformation is essentially subjective, and one which we should not expect to find consolidated at implementation level. Nothing in Boden's example set of systems, or in our understanding of how those systems work, suggests a concrete necessity for it. And the observation that any transformational process must operate, ultimately, on a structure in which conceptual spaces are not explicitly represented suggests, rather, that transformation has to be reducible to the more mundane process of conceptual search. Finally, we also have Wiggins formal demonstration that the two processes are effectively equivalent [Wiggins, 2001].
As an initial venture, we might try substituting the complexity measure (Equation 1) directly into the definition, as a self-contained replacement for the all-or-nothing criterion.
Visual illustration of conceptual complexity.
Boden herself has commented on the general idea of using a complexity rule as a means of evaluating creativity, pointing out specific requirements that any satisfactory measure would have to satisfy. In discussing the general question of whether creativity can be measured she writes `Some form of complexity measurement, as used by computer scientists, would be useful. However, depth within the space must be recognised too. The appropriate method of assessment would have to take into account the fact that conceptual spaces are multidimensional structures, where some features are ``deeper'', more influential, than others.' [Boden, 1994a, p. 113]
She goes on to say that a satisfactory measure should somehow return a `structured distance' and be sensitive to `interesting differences' between artefacts of different styles or genres.((She gives the example of a comparison of the degree of creativity in the Mona Lisa as opposed to the Demoiselles d'Avignon.)) It should somehow reconcile different dimensions of creativity and deal properly with the tradeoff between conceptual `depth' and `number'. It should also properly relate transformational creativity to exploratory creativity. [ibid., p. 114-115] How well does the proposed definition stand up to these demands?
Because of the way the growth formula generalises exploratory and transformational processes, it is capable of capturing the differences between these two modes of operation without modification. It also handles the tradeoff between conceptual `depth' and `number' in the sense it measures both things in the same way---in terms of the amount of work involved in the generation of the relevant concept.
The problem relating to incommensurability among concepts, is also solved but with a caveat. On the assumption that different genres embody concepts drawn from different conceptual spaces, no space-specific measure can deliver an unambiguous cross-genre comparison. However, measures produced this way are commensurable in the sense that they are calculated in the same units. Thus there is the potential for cross-genre comparison provided that the relative contribution embodied in the respective primitives is taken into account.
Overall, the replacement of the all-or-nothing criterion with the complexity measure produces a revised definition which implements Bundy's suggestion reasonably well, and does so in a manner which satisfies most of Boden's requirements (always remembering that these were not actually intended for this use). But it seems to me that, even so, it is not quite what is wanted. Something important from the original formulation is being lost.
Although the systems evidence Boden brings to bear may not quite compel us to believe in the mediational role of conceptual-space structures, the human evidence is more persuasive. The anecdotal evidence Boden assembles appears to suggest it is key facet of creativity that it is mediated within structures or frameworks of concepts, or theories. For a satisfactory re-working of the definition, this needs to be brought out. What is really needed is a complexity-based definition which somehow gives the conceptual space the prominent position it formerly occupied.
To calculate how much this is, we only have to look at the extension or coverage of the relevant concept. The establishment of a concept covering all the entities in a particular set X means that they may subsequently be processed as a single entity. That is, after all, the key goal of conceptualisation. The representational gain may then be measured in terms of the difference between having to process all members of X individually and being able to process them as a single entity. In other words, the representation gain may be measured in terms of the size of X.
This leads directly to the following formulation for net conceptual complexity:
is the
complexity of concept c, derived using Equation 1 with
. 
is the size of the extension of
concept c. The formula asserts that the net conceptual
complexity (NCC) of concept c derived in a hierarchy based on
primitives P is the objective conceptual complexity less the
size of c's extension. A schematic illustration appears in
figure-below.
Visual illustration of net conceptual complexity.
The NCC seems to be a better basis on which to ground the definition, then, since it allows us to have our cake and eat it too with respect to grain-size. Should we prefer the all-or-nothing approach, we can achieve this by the identification of `radical creativity' with the production of negative NCC values. Should we prefer the graded view, we can achieve this by using NCC values directly.
In addition, utilisation of NCC as a foundation for the definition allows the incorporation of an evaluative element. The original revision, like Boden's criterion itself, ignores the value of conceptualisation, focussing exclusively on the amount and nature of work involved. On this type of view, a useful concept may be rated as being just as `creative' as a useless concept, provided only that the conceptualisation effort is equivalent.
Using the concept of NCC within the definition allows us to get beyond this, at least with respect to contexts in which representational power is significant.((Presumably these are the ones where creativity is more intellectual than artistic.)) It allows the definition to capture not only the idea that creativity involves representational work at a specific level, but also the idea that it should involve movement in the direction of greater representational coverage.
With this labelling in mind, it is possible to view the conceptual universe as a landscape of peaks and troughs. Peaks occur in the vicinity of concepts with high NCC, small elevations where concepts show low NCC, and where concepts show negative NCC, there are marked depressions. The landscape thus shows regions of high, low and negative NCC.
In the case where a set of concepts form the primitive constituents of an explanatory theory, we would expect to find a marked depression `upstream' in the hierarchy. This depression will be made up of all the concepts which utilise the theory and which share in its explanatory power. The depression is a zone in which conceptualisation has the effect of reducing conceptual complexity. Putting it another way, the depression acts like a `sponge', soaking up and eliminating complexity from the concept hierarchy. See figure-below for an illustration.
Sponge zones in a conceptual space.
Since every conceptual space is based on primitives which, by definition, provide the basis of an explanatory framework or theory, there is a direct correlation between conceptual spaces and sponge zones. Every conceptual space is a sponge zone. The explanatory power of the space corresponding to the `absorbency' of the sponge, i.e., its capacity to eliminate complexity.
Boden's observations about the ways in which creative thinkers operate with respect to conceptual spaces can thus be understood in terms of sponge zones and complexity reduction. Exploration is the attempt to focus conceptualisation within zones of maximum dividend. Transformation can be understood as the attempt to open up new zones, in which exploration may then be profitably applied.
Recognition of the role played by sponge zones allows us to see that this approach in no way undermines Boden's usage of the conceptual space as an explanatory entity. Rather it supports and confirms it by illuminating the relevant efficiency implications. The original account, with the NCC-based definitions substituted for the all-or-nothing criterion, would therefore appear to form a valid extension of her theory.
Boden herself might not be completely happy with it, however. She has registered some resistance to the idea of a complexity-based definition on the grounds that it could not truly reflect the multi-dimensional nature of creativity. As she comments in a response to a reviewer, `I prefer to avoid speaking of the ``degree'' of creativity, since to do so implies a continuous spectrum along which individual thoughts are to be ordered. To the contrary, a main theme of [The Creative Mind] is that creativity is multi-dimensional.' [Boden, 1995, p. 169] To me this seems unnecessarily restrictive. We regularly use 1-dimensional measures of multi-dimensional phenomena. Indeed we do so every time we speak about distance in 3-dimensional space. So I feel we should not necessarily baulk at the idea of a definition based on a 1-dimensional measure.((Were we to do so, we should arguably have the same reaction to the original definition.))
Others may find the revision more appealing since it does appear to answer some of the criticisms which have been raised. Indeed, it would appear to answer Ram et al.'s requirements [Ram et al. 1995, p. 122] directly. Though quite happy with the overall form of the account, these reviewers worry that it does not sufficiently clarify the way transformation is grounded in normal conceptual operations. As they note, `The question is though how the search space comes to be expanded to facilitate creative thought using ordinary mechanisms.' In particular they want to know how `ordinary operators and processes can take the reasoner out of the space that would usually be explored.' [ibid, p. 122]
Since it takes account of the reducibility of transformation to ordinary conceptual search, the revision is able to answer this question while still staying within the general scope of the original account. Ram el al. should therefore find it acceptable. As they note `we would prefer a model of long-term conceptual development in which the individual evolves a search space, that, when explored by normal thought processes, still includes many thoughts that would be considered creative. [ibid, p. 114] The revision delivers precisely this enhancement.
An ideal definition of creativity would be fully objective, of course. But this could only be achieved by establishing a fixed starting point for conceptual development, i.e., a `standard conceptual agent' equipped with a specific set of primitive concepts and constructive operations. On this basis, it would then be possible to measure the creativity of any concept against a fixed background, perhaps producing values expressed in thought years (analogous to light years) which would be the amount of work required from the standard agent in order to produce the relevant concept. (figure-below provides a schematic illustration of the way this might work with respect to some financial concepts.)
Imaginary thought-year distances for some financial concepts.
One issue on which the two accounts are already in complete agreement is the question of `special powers.' In Boden's view, it was a key strength of Koestler's approach that it `appealed to no special creative faculty, granted only to an elite.' [Boden, 1990, p. 24]. Boden's own theory closely follows Koestler's line on this, stressing that `creativity is not a single ability, or talent.' [ibid., p. 12] Both writers, then, see creativity as a process closely integrated within other processes of cognition.
The revision does not in any way contradict this `diffusional' view. Rather it consolidates it in the form of a still more radical proposition: that as a process, creativity is actually indistinguishable from ordinary conceptual development. In other words, there is no substantive difference to be discerned in the underlying mechanisms of conceptual development, and those of creativity. Of course, as a result of having different experiences, and perhaps of being equipped with a different set of original concepts, different concept-using agents will inevitably produce different results. But, on the present account, the deed is done in the same way.
This idea encourages an egalitarian viewpoint in which `creative masters' are understood to be using the same mental equipment as `common-or-garden thinkers'. While they may produce very different results, this would seem to be only because of different experiences and starting conditions. On this view, Kekule's dream-inspired innovation of the benzene ring concept was achieved by exactly the same process as would have been involved in the imaginary crossing-sweeper's innovation of the `nicely swept crossing' concept. Indeed, if we believe that animals generate concepts in the same way we do, we might have to go further still, perhaps accepting that Kekule's ring concept was derived by the same mechanism as that used by, say, a large labrador developing the concept of `small-furry-chasable-thing'. In the context of science, one innovation may be of greater magnificence than another. In the context of mechanism, there may, after all, be very little difference.
The complexity formula for conceptualisation was utilised for purposes of constructing a revised definition of creativity. The final version of this defines creativity in terms of net conceptual complexity, with the category of `radical creativity' being reserved for concepts carrying negative NCC. The revision was found to answer requests for a theory which would clarify the basis of creativity in `ordinary thought' and to deal more directly with the question of evaluation. Finally, it was found to add additional stress to some of the fundamental themes of Boden's account, including the insistence that creativity should be regarded as an integral part of ordinary cognition. In sum, the revision bears out much of what Boden has argued, but allows it to rest, definitionally-speaking, on firmer ground.
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