Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003, Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
NESTED, CHAINED AND INTERSECTING
or ‘COMPLEX’ CONTRADICTIONS
Anja-Karina Pahl and Olga Bogatyreva
Centre for Biomimetics and Natural Technologies,
The University of Bath, BA1 2EU, UK
Tel +44 (0) 1225 826 956 Email
Abstract
The mathematical structure and representation of chains, nesting and intersection is described in relation to contradiction in TRIZ. Each of these types of contradiction is related to the others, in 3D, in a conic representation. Complex contradictions are introduced as those involving two sets of distinctly different original contradictions or cones. Study of ant behaviour provides some insight into solving complex contradiction –in particular, contradictions need not be solved in the same sphere in which the problem is caused. Penultimately, the solution to complex contradiction exists in a concept space, unoccupied by previous solutions. And by involving other spheres of information to solve complex contradiction, all contingent and relatively simple contradictions should also automatically be solved.
1.0 INTRODUCTION
In complex systems, it is quite common that contradictions, which exist at one level of operation, or in one sphere of activity, can also contradict each other at another level. Then we have a complex situation of ‘contradiction between contradictions’. What is more, our engineering problem may require us to find solutions to all of them.
Ideally, all relevant, potentially multiple contradictions should be resolved simultaneously. In other words, rather than identifying a separate but apparently ‘real’ space-time for each situation, we identify that they occupy the same conceptual space-time and solve all ‘paired’ contradictions together.
There are several types of complex contradictions. While ‘chained contradictions’ have been introduced by Mann (2000), and the concept of nested solutions has been introduced by Pahl (2002), TRIZ has as yet no directions for solving these or any other type of intersecting contradiction.
In this paper therefore, we will review the mathematical structure and representation of chains, nesting and intersection, and how this is relevant to describing problems and solutions in TRIZ. Then we will investigate how complex contradictions have been solved in a natural situation, greatly aiding our process of doing the same in TRIZ.
1.1 Line of Contradiction
Figure 1 The simplest description of contradiction
is a line that does not involve the resolution.
To describe the simplest case of a contradiction, we might say that it is graphically represented by a line or curve with two end points –which symbolize the extreme targets, goals or conflicting parameters that must be integrated (Fig.1). In this representation, the line of contradiction is independent of the resolution.
1.2 Chained Contradiction
The next simplest representation, as discussed by Mann (2000), is a ‘chain’ (Fig. 2), where each successive solution moves closer to the optimum.
Said chain evolves since solutions to technical contradictions are not always discrete and do not always immediately eliminate the entire contradiction –instead they often merely improve a situation involving conflict. Then, because we have not achieved the ideal solution, but only a partial one, we must continue to either solve new problems for the old constraints (Mann, ibid), or, as we argue in this paper, discover yet more contradictions with different constraints.
In Mann’s (ibid) chain, the contradictions are not obviously related to each other (cf. Figure 1.) –they are only related to the solution. Thus the chain is, in fact, not a chain of continuous contradiction, but a scenario of continual resolution.
Further, contrary to Mann’s (ibid) assertion, a paradigm shift can never really happen, because the entire journey takes place for the same initial set of conflicting parameters.
In order to initiate a true paradigm shift, we need to involve a second, third or nth chain. This means we discover, or are presented with, new sets of contradictions or entirely new end-member parameters, which do not fit into the existing system. When this additional information or new criteria is made relevant to the original complexity, it also becomes possible to achieve an overview of our situation. This then, is where it becomes possible to initiate a shift in our thinking (Fig.3). This concept is extended in the next section.
1.3 Nested Contradiction
The next simplest representation of contradiction is as a triangle.
As Pahl (2002) pointed out, conflicting or contradictory parameters for a given problem are not necessarily anchored or anchorable, but can exist anywhere along a line of contradiction. In the same way, the actual solution is not originally, or usually, a fixed point or target, but can occur anywhere along a sliding scale that is also represented by a curve or line –this time at right angles to the first.
A triangle will be created by linking all extremes –that is, linking the (line of) conflicting parameters with the (chain to ideal) Final Result (Fig. 4).
Of course, potentially infinite triangles may be inscribed within this ideal triangle, as a result of having one, two or three sides shorter than the ideal.
When these have regular sides and occur in a sequence, as the contradiction set and/or solution is systematically varied, they will appear to the eye to be ‘nested’. There are three types of regular nesting explained in figure 4, the most obvious of which is in the manner of a matriushka doll.
Strictly speaking, irregulartriangles created within the meta-triangle, are also nested. These will be overlapping or intersecting, depending on the degree to which they are anchored on (or share) a contradiction pair or solution (fig.5).
AB
C
This inclusive view of a complex situation actually extends the concept of chained contradiction presented above, in that all chains or bifurcated lines within a meta-triangle must eventually be related (fig.6).
Actually, all three methods of representing contradiction, which are discussed in this paper, are related. For, as Pahl (2002a,b) has also shown, triangles are a 2D representation of the path of conical spirals or volume of cones, whose mouth is a circle or ellipse. Thus we can also imagine contradiction not in exact lines or triangles, but as circles enclosing more diffuse sets of potential solutions (fig.7).
Intersecting contradictions are for this reason discussed following.
AB
C
1.4 Intersected Contradictions
As we know, 1D lines (which create chains) are simpler mathematical constructs than 2D areas (for example, triangles or circles), which are themselves simpler than 3D volumes (created by conical or vortical spirals). Thus the least simple representation of TRIZ contradiction involves circular sets and 3D cones.
Mathematically, we can calculate the precise point where an intersection of two curves or single lines occurs, by using the ‘simultaneous equation’ method of equating equations. We can do a similar exercise to find the overlap of two areas (fig.8).
Intersecting Contradiction
A
The advantage of looking at contradictions in this way, is that we see that we can combine two or more completely different spheres of activity (the individual meta-triangles or chains of previous sections), each of which have their own internal constraints and contradictions. Thus the real 3D picture of complex contradiction is described in Figure 9.
SOLVING INTERSECTED CONTRADICTION
Now we will look at how nature solves intersected contradictions.
2.1 Advice from Ants
The ideal result for any society is ‘cooperation’, whereby the society acts as a unified organism and survives in equilibrium with the environment. However, at the same time as satisfying society, there are conflicting demands on individuals. Each member wants also to expend as little energy as possible. In other words, the colony demands working elements, but nobody wants to work –a very typical situation for any society!
For the ant society, we can cite two main pairs of contradictions, which must be solved together: specialization-generalization and cooperation-competition. We can formally state that the main contradiction is to ‘minimise personal use of energy and maximise reliability of the colony’. Because the pairs of contradictions involve different spheres of activity, the simultaneous solving of both (ie. the pairing of pairs) is a complex contradiction, not a simple one. Graphically, we have two intersecting cones or circles, with different constraints (fig.10).
In Table 1., all possible solutions of all simple and complex contradictions for this situation are presented (Bogatyreva, 2002). The type of response, which we might call the ‘solution’ or ‘management’ of ants, is divided into four spheres of activity, which for convenience we call Activation, Controlling, Coordination, Governing. These are different levels of creating cooperation, which are brought into play as the colony evolves.
Figure 10 Complex contradiction in an ant nest involves two sets or pairs of criteria which must be simultaneously resolved. As shown in Figure 8., there are at least two distinct solutions, mathematically defined by the points of intersection, or overlap.
In a young colony, or when an individual is young, each ant can theoretically perform any task, though it is (somehow) activated to choose and do only one task at a given time. At such an early stage, expertise is so low and competition between individuals so intense, that there are no stable interactions. However, the more active ants guide others in a task. They are the activators. Once an ant has chosen its task, this determines which group it can join –ants band together to perform a similar or identical function, and can help each other, or cooperate in such teams. Because the level of function is still quite simple within a team, this requires little effort in coordination between individuals –it is fairly automatic. Yet at the same time, there may be an ant who has control over ‘membership’ of a team, selecting only fit and able individuals. Next, as the number of ants in a colony increases, the number of functions that can be performed increases –which means the number of teams that form to carry out these functions increases. This now requires coordination of action between teams, a higher level of complexity – thus one ant will take on the task of making sure the team can cooperate with others in every necessary way. Last, there is coordination of all the teams in the nest, carried out by an individual who has in its lifetime performed many functions, and can now sit ‘outside the nest’ and regulate it or govern.
Table 1: Types of “management” employed by ants to solve simple and complex contradictions in four spheres of activity
Contradictions / Sphere of activitySpace
(fixed material boundaries) / Time
(fixed seasonal and diurnal constraints) / Structure/ Function (dynamic individual response) / Interaction (dynamic communal response)
Simple / 1a Specialization /
III
1b Generalization2a Cooperation / II
2b Competition
Complex / Any combination of 1 and 2 / IV / I
Bogatyreva’s (2002) paper indicates that contradictions such as ‘cooperation-competition’ can be solved simply by controlling (II), and in the complex sense by activation (I), both of which call forth specialised behaviour –meaning that someone who likes doing something will attract and activate friends for this particular job.
Likewise, the contradiction ‘specialization-generalization’ can be solved simply by coordination (III) – inducing a strong behavioural hierarchy. Competition between individuals is hereby effectively ‘frozen’ in the interaction structure –there is just no point in fighting every time, if all leaders are recognisable. Governing (IV) is a way of solving a complex contradiction between individual and social spheres, by locating a central government for outdoor work in a nest.
There are additional considerations, which determine how a contradiction is solved. For example, biologically, colony “management” depends both on size of population and complexity of society. Ants with a small population demonstrate Activation (I) as the way of solving complex contradiction. Ants with a complex social structure solve the same complex contradiction by Governing (IV).
2.2 Degree of contradiction
We know from earlier in this paper, that we can geometrically relate cones and circles with triangles and lines. We thus also know we can eventually relate all chains, nests and sets of contradiction.
That means there is no absolute measure of the degree of contradiction. Essentially, whatever the starting point is, where our initial contradictions are defined, must be our first level or degree of contradiction. Subsequent contradictions identified at the next stage of problem-solving will be second degree or level 2, thereafter third degree, and so on. Each is countable only in a chain as consequent, or contingent on, an intial or prior pair of contradictions and within the same overall set.
To some extent it will be self-evident that the degree or level of complexity of contradiction is equivalent to the degree to which a given structure (ie. set of contradictions) is different from a second or third set, since this will require more connections to be made between chains of information or contradiction, before sets can be adequately related. However, it is unlikely that this will be obvious at the starting point.
In other words, we define a complex contradiction as having constraints in two different sets of spheres of action, which are not contingent on each other, at the level of our observation.
In any case, at each stage of solving a problem, we must ask these sorts of questions: How is one set of contradictions related to another set of contradictions? Do we have (two or more) completely different cones/triangles/ circles/end-member criteria, or not? In other words, are sets of contradictions truly independent at each new starting point, or are they contingent on each other?
As corollary, from Bogatyreva’s (2002) research, we can add that a simple contradiction will be solved close to the original sphere of influence. Thus, a simple contradiction in relationship is solved in the sphere of the dynamic response of the individual (Structure/Function). A simple contradiction in individual function can be solved purely by dynamic community response (Interactions sphere).
On the other hand, solutions of complex contradictions lie outside the original areas of influence. For ants these solutions lie in what Bogatyreva (2002) calls the spheres of ‘Space’ and ‘Time’. These are domains of action, which may originally have formed in response to demands of the society –for example, the spatial limit of the nest is a function of the size of the colony. Similarly, the division of a workforce into simultaneously labouring teams (creating ‘work divided in time’), also depends on the size of the colony. However, the constraints these spheres later impose on both the individual and the community become anchored.
3.0 DISCUSSION
Now we will discuss whether intersected contradictions are a peculiarity of complex living systems, or whether we can find them in thinking techniques too.
Let’s have a look at typical TRIZ task, described in many Altshuller books. The task is about hot liquid slag in a bucket, which needs to be transported from the blast furnace to the place where it will be used.
While transporting it, a hard crust appears on the slag surface. To remove the slag from the bucket, it will later be necessary to break the crust with a special device. But the crust is thick and the shape of bucket is conical, which means that even when the liquid is poured out, lots of hard slag will be left rimming the bucket. An additional procedure for breaking and removing the hard slag is required. The problem is how to avoid loss of slag at the same time as minimizing additional procedures.
We test the solutions made available by understanding ant colony organization above.
Basically, there are two functions to be improved – time of slag transportation and slag insulation. These create two simple contradictions – a time contradiction (transport / do not transport) and a structure contradiction (insulate / do not insulate). The simplest decisions, and the most desirable ones, are to transport and let slag harden (do not insulate) and they form a complex contradiction with the goal (ideal result) of the system, which is to empty the bucket easily and without loss of slag.
We may wish to introduce a metal bucket lid, as a new structure in this system. This will help stop spillage during transport. But will it help or hinder the slag hardening?
Table 2: Solution table for applying complex contradiction
in ant behaviour to a problem with slag
Contradiction / DecisionsSpace / Time / Structure/
Function / Interaction
SIMPLE
Convert quickly /
- Transport
- Do not transport
Convert after transporting
SIMPLE
Harden (slag) /
- To delay hardening, lid should exist.
- To let slag harden, lid should not exist
Do not harden
COMPLEX
Transport / Add ice as new structure in system, to make an insulating foam crust, that is easily removed
Do not insulate (let slag harden)
If we consider the solutions from ant organization, proposed in this paper, a simple time contradiction should be solved in space while a simple structure/function contradiction is best solved in time. We should also try to make decisions or find solutions in the structure/function and interactions spheres. The best decision regarding the complex contradiction ‘transport /do not insulate’ is in Structure/Function sphere.
Thus the ant’s complex contradiction rule works in our technical system too. As the system described is very simple, there is no second way of solving the complex contradiction.
4.0 CONCLUSION
In this paper we have proposed that, when solving simple and complex problems, different types of contradictions will become evident. Depending on the degree of complexity of problem, chained, nested or intersected contradictions can be identified. Ultimately, these are related to each other –geometrically the situation is represented in 3D cones.