Contradiction and Analogy as the Basis for Inventive Thinking

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Anja-Karina Pahl
Department of Mechanical Engineering, University of Bath, UK
ensakp@bath.ac.uk

In order to find one’s place in the infinitude of being, one must be able to both separate and unite (i)…

In this paper, I explore the nature of contradiction and analogy as a basis for inventive thinking. In doing so, I find that contradiction is not only a primary constituent of TRIZ, but fundamental to creating scale, perspective and orientation -that is all the ‘dimensions’ perceived in our mathematics, neurophysiology and, indeed, the physical world. Contradiction is likewise fundamental to the process of determining analogy -or creating fields of relationship, in technical innovation. Both analogy and contradiction are most easily represented by relating elements in a triangle, reminiscent of the relationships created in Su-field models. There may be many nested levels of triangular relationship required to describe an integrated system, sub- and super- system during its evolution from past to future. Both the presence of analogy and resolution of contradiction force us to identify the system level within which we function, and from which we must remove ourselves to apply a domain-specific idea at another level, or in another context. On the basis of both mathematics and intuition, contradiction is intrinsic to multi-dimensional, analogic and inventive thinking.

In Western science, the formative years of our training are traditionally based on Newtonian abstraction. The methods we are taught do not make us comfortable with relating to, or resolving, contradictory information or modes of thought. They teach us, instead, to screen out one of the options. They force us to eradicate apparently extraneous data and opinion, in the interests of over-simplicity, predictability and economic or temporal efficiency, just as soon as they seem irrelevant to our original aim.

The fact is, however, that in this way, we often throw our proverbial baby out with the claw-footed bath. First, because abstracted thinking is inappropriate for dealing with a natural world of chaotic, dynamic and complex behaviours. Second, because, as modern psychologists Howard Gardner [1, 2] and Bernice McCarthy [3, 4] point out, the acknowledgement of contradictory information is an important part of the rational learning process, the creative process and the evolution of integrated intelligence.

TRIZ has become a highly popular aid for creating innovative technologies in the West. Compared with other ‘thinking systems’ used commercially, it delivers considerably more complex tools and information. It also allows for both rational and intuitive thought. The fact that it does so successfully is likely due to an internal simplicity -the suite of TRIZ tools is permeated by two outstanding features:

In this paper, we will explore how these two factors are intimately related and may fundamentally underlie both our perception of the existing world and our creative construction of new ones. There are three main questions we will address:

The Oxford and Macquarie dictionaries [5, 6] define contradiction as, ‘a lack of agreement between facts, opinions, actions etc.’ From Roget’s Thesaurus [7], we can also add the synonyms; ‘being contrary; a contrast, dissent, discordance, difference, diversity, discrepancy, incompatibility, opposite, antithesis.’

That said, what absolutely constitutes contradiction is, of course, a matter of opinion.

The lack of agreement between observers/observations is totally a function of their respective starting positions (both conceptual axioms and assumptions or physical position and orientation) with respect to the object of interest. The discrepancies in their data are likely scale-dependent. Dissent may exist only in the reference frame within which one or other observer primarily operates -for we can certainly find a relationship of elements within contradiction, which simultaneously separates and unites concepts. And we can sanely redefine our starting points, to find that the contradictions were only apparent, and not ‘real’ - i.e. a paradox.

TRIZ defines two types of contradiction -technical and physical. The first (eg. desiring both A + B) can be considered ‘conceptual’ or ‘apparent’, and is presumed to be resolvable, by considering the problem from outside the existing system (using one of the given tools). The second (eg. desiring A + not A) is usually considered absolutely ‘real’ or ‘actual’, and unable to ever be resolved. Of course, in the terms just discussed, this may also not be true. The definition of contradiction as ‘antithesis’ will allow us to get rid of personal opinions as to what contradiction might be.

In any case, as Nikolay Shpakovsky [8] also pointed out, we can see the resolution of antitheses very clearly and simply in a triangle (Figure 1), where two extreme positions during the problem-solving process are ultimately united in a third position -the final or ideal result (IFR). At the apex of the diagram, the distinct situational levels of object, concept and abstraction are merged. Along the way to resolution, there is some conceptual ‘to-ing and fro-ing’ between endpoints, over the central axis, creating a kind of ‘Christmas tree’ effect. And, of course, the process of resolving discrepancies may stop before reaching the ideal endpoint. However, that will not change the shape of the relationship -merely leave us in a ‘nested’ tree, or sub-system level solution, a situation to be further explored below.

TRIZ presupposes that, ‘whatever your problem is, someone, somewhere, has already solved it’, and that we can draw on the experience of others and a database of solutions approximately, if not exactly, the same as our own, through analogously applying the 40 Principles of Invention to our problem system.

Figure 1. ‘Christmas tree’ diagram, after Shpakovsky. Horizontal axis is degree of abstraction of the situation. Vertical axis is ideality of solution concepts. Points along this axis correspond to situations in which contradiction appears to exist, because concept and physical object levels are mixed. The problem-solving process involves many transitions between expert competence with object knowledge, and specialist competence with abstract knowledge.

Analogy is recognized as perhaps one of the most important teaching methods, especially where some linguistic difference exists between speaker and listener. As a symbolic representation, it is natural to many cultures, important as a means of popularizing or guiding complex, abstract concepts. It creates a shortcut or ‘fast-track’ to comprehension, as it offers the advantage of drawing on some pre-existing capacity to see similarities and differences, rather than assuming a lengthy education is necessary. It may have a central role underlying the architecture of all thought [9].

The world’s great scientists have recognized analogy as being important in their research and variously labelled it as: a useful story, hypothesis, assumption, model, theory, standard, proof, reference frame or framework [10]. The Collins and Macquarie dictionaries describe analogy as a ‘plan, blueprint, template, definition, concept, representation’ [ibid]. The Encyclopaedia of World Problems and Human Potential adds to this list with; allegory, synecdoche, metonymy, parable, symbol [11].

In short, the modern Western world regards analogy as an abstraction; a vague aesthetic, linguistic, symbolic, visual or mechanistic approximation. It supposes analogy is a ‘state of being something’, occurring at a single point in spacetime. It demands analogy to be a singular event, mental state or observation, even though this greatly limits its usefulness (please see Appendix for further information).

In the strictest sense however, analogy is not any of these things.

Analogy is a mathematical relationship, responsible for the repetition of individual and group elements in natural and artificially produced systems. It is the means whereby music, art and architecture achieve their precision and complexity. Related to stereology and also known as projective geometry, it is classically a technique whereby all conceivable 2D and 3D geometries can be compared at a single scale of observation: actually distorted in 2D, in order to be apparently preserved.

Classically, according to the ancient Greek philosophers, analogy does not just ‘exist’. It is formed. Analogy is a process, achieved via a certain sequence of steps [12].

These are:

The final proportion requires the largest term to be a wholeness or unit, composed of the other two terms -ie. a repetition of pattern involving at least two scales. It should be obvious that growth is here achieved simultaneously by addition, multiplication and division, in a sequence or series.

Two interesting correspondences are implicit in the classical process of analogy, which are potentially important for our understanding of its function in TRIZ and inventive thinking in general.

These are:

• When we create a ratio, we are comparing or examining differences of entities assumed to be largely the same. That is, we are comparing magnitude, though not identity. Then, determining ‘differentness’ can be made more exacting through determining ‘sameness’ - that is, by reducing the number of variables in the equation. The process thus first requires objects to exist as isolated entities in their own, local, frames of reference, somewhere apart from a hidden observer, who is also in her own local frame of reference. Then it requires that all reference frames, including that of the observer, be related to each other. In other words, as the steps of analogy are followed and resolution is increased, the observer both becomes explicit and is built into the system. In the final, continuous and golden steps, where we are using the fewest terms or objects for comparison, it is the perceiver herself, b, who consciously integrates or forms the equivalence and identity between the observed differences. That means all previous mismatches, discrepancies or apparent contradictions in our thinking should have been resolved at this stage.
   
• The whole process of establishing analogy is fractal or scale-independent, since the reference frame is complete at each stage yet is simultaneously additive, multiplicative and divisive, as new elements are introduced and made consistent with the primary observation. At all points in the process of creating analogy, the system involves more than one scale. The penultimate system includes all scales. In other words, there are hidden ‘dimensions’ and related system levels at every stage of both time and space. This concept can perhaps be further exploited in the context of TRIZ ‘system operator’ or ‘9-windows’ [13]. We must assume that there are hidden but related systems in our problems, at every stage, even when we cannot immediately see them.

Interestingly, classical analogic thinking is creeping into the modern study of chaos and complexity. Creations like ‘Pseudo Phase Space’, for instance, involve only one variable changing in time [14]. First, new variables are created by comparing the original identity with a time-lagged identity and then both starting and finishing points are tracked as they create sets, (sub)series or sequences of ratios. eg. xt: xt+1, xt: xt+2, xt: xt+w… Similarly, there exist Non-integer or Exponent -so-called ‘Fractal Dimensions’, which relate the number of increments needed to measure an object, to its evolution. The variables are thus directly proportional to the size of the scale or measurement tool and come close to including the observer in the system [15].

Figure 2. Nested Triangle diagram.. Horizontal axis is degree of contradiction of the situation. Vertical axis is ideality of solution concepts. Points along this axis correspond to situations in which contradiction is resolved, as if the result were final or ideal. It is however, possible to achieve ever-increasing levels of resolution, hence we can identify or track relative subsystem, system and supersystem levels. Again, the problem-solving process involves many transitions back and forth between concepts or parameters, considering fully first one side of the argument, then the next.

Again, we can illustrate these concepts of analogy very clearly and simply in a series of nested triangles (Figure 2), of intermediate solutions (labelled relatively as subsystem, system, supersystem), where two extreme positions are ultimately united in an ideal situation -the IFR. Note that the axis of ideality is also an axis of time. In contrast to Shpakovsky’s model, contradiction does not exist as a point in space, but as a line between conflicting parameters. Accepting that contradiction is therefore our very journey back and forth, we can be, in this case, perhaps more fully engaged with the process of resolution than when we start from the divisive standpoint in which contradiction exists as a static entity. Necessarily, before resolution, there may still be some conceptual ‘to-ing and fro-ing’ and different levels of solution which are not ideal (although often, in some sense, final).

A similar model of triangular relationship is necessary to understand depth and perspective.

Neurophysiologically and mathematically, it turns out that depth is an attempt to symmetrically resolve a mismatch of 2D information. It is not always necessarily a measure of ‘real’ or physical space.

We have, for instance, two eyes, rather than one cyclopian one, so that slightly different orientations of an object are presented to each eye [16]. As the brain tries to find a relationship between them, we interpret their difference as a third spatial dimension or depth (Figure 3a).

Now, in the ideal situation, where an object is located at the centre line of vision, we would say that depth of data is the common side of two identical, symmetric triangles sharing a common, primary reference frame. The situation gets trickier when we move off the centre line, for that introduces at least two more (local or non-primary) frames of reference (one for each eye), which are either slightly rotated or at different scales, with respect to each other (Figure 3b). In other words, the system evolves, each time an element moves… we get a new set of contradictions and triangles and quasi Su-field relationships*.

Objects located off the centre line of our binocular vision always have one object-to-eye path, which is slightly shorter than the other and which will take less time to complete [17, 18, 19]. And there are two ways in which we can resolve the discrepancy in object-to-eye distance:

(I) If we preserve symmetry, there are two possible positions for the object to occupy; O1 and O2, which create the same total discrepancy in object-to-eye distance and whereby the angles of rotation will cancel out. This will create four local frames of reference.

(II) If we account for scale, there is a single reference frame on one of the object positions, which can harmonize the discrepant angles via two similar triangles: RF1B and RF2B sharing a common side.

We exploit this phenomenon in modern computer graphics, by encoding left-eye and right-eye images on alternate fields, and juxtaposing these temporally on our monitors. The faster this happens, the easier it is to sustain an image, so the fastest machines today change image direction approximately 120 times per second.

Ears, analogously, find ‘timbre’ and ‘beat frequencies’ in music, via superposition of dissimilar sound waves received in stereo [20]. Depth, in other words, can result from temporal mismatches as well as spatial ones.

We can also presume that the ability to find depth, or additional meaningfully related information, in a given philosophy or theory (such as TRIZ) follows the same rules. Deep understanding of a subject usually follows from resolving apparently conflicting information and finding a relationship between elements that creates some useful analogy.

(a)	(b)

Figure 3. In binocular vision, depth is created by temporal or spatial differences in arrival of information. In an ideal situation (a), with an object located in the centre line of vision, depth is the third side of two similar triangles, sharing a common or primary reference frame (PRF). In a non-ideal situation (b), with an object located anywhere off the centre line, the distance from eye to object is longer for one eye than for another. There are now two local frames of reference (RF1 and RF2) rotated with respect to each other. Note, however, that symmetry can be preserved (i) because there are two possible positions for the object to occupy, with the same difference in object to eye distance (O1 and O2), and (ii) also because we can resolve differences in angle by accounting for changes in scale (RF1b and RF2b).

The ability to see a given subject from different points of view is linguistically referred to as ‘having perspective’. In both engineering and daily life, we know that being able to view a problematic engine or tactical dilemma from various angles often leads to unique insights and solutions.

We can treat this issue as a corollary or extension of depth.

It was the great painters of the Renaissance, who developed the system of ‘focused perspective’, which depicted objects as they appeared to the eye. Their discovery heralded a significant departure from the traditionally ‘flat’ art and thinking of the Middle and Dark Ages.

The fundamental principle of perspective is the projection of multi-dimensional objects onto a lower-dimensional -usually 2D plane, through a given focus. It visually preserves all angular relationships of elements within the original nD object, though usually this necessitates an actual distortion of shape on the (n-1 D) page, in order for the visual effect to remain true [21, 22] In chaos theory, we say that the different viewpoints remain topologically equivalent. But this means the visual geometry requires that volumes shrink with distance and parallel lines converge to a vanishing point on the horizon. This simplest example of this principle is parallel railway lines converging to a point (Figure 4a).

A slightly more complex example in which distortion of a shape conserves visual geometry is Leon Batista Alberti’s (c1440) outline of the general method to represent horizontal squares in a vertical picture plane as trapezoids [23] (Figure 4b).

(a)	(b)

Figure 4. (a) In order to conserve visual geometry on paper, we apply perspective such that parallel lines must converge to vanishing points. (b) see body of paper for description.

Let the eye be at a station point, S, that is h units above the ground plane and k units in front of the picture plane. The intersection of the ground plane and the picture plane is called the ground-line; the foot, V, of the perpendicular from S to the picture plane is called the centre of vision or vanishing point; the line through V parallel to the ground-line is known as the vanishing line and the points P and Q on this line are called the distance points. If we take the points A, B, C, D, E, F, G marking equal distances along the ground-line RT, where D is the intersection of this line with the vertical plane through S and V, and if we draw lines connecting these points with V, then the projection of these last lines, with S as a centre upon the ground plane will be a set of parallel and equidistant lines. If P (or Q) is connected with the point B, C, D, E, F, G to form another set of lines intersecting AV in points H, I, J, K, L, M and if through the latter points parallel are drawn to the ground-line RT, then the set of trapezoids in the picture plane will correspond to a set of squares in the ground plane.

Perspective requires increasingly sophisticated methods of geometrical analysis, involving not only straight lines, but conics. It means we coincidentally apply the principle of scale in order to depict object orientation and depth accurately.

In the TRIZ framework, we might say that once we have established how a problem or object, with its local reference frame, sits in our primary reference frame, ie. in relation to ourselves, the observer at the origin, then we can then rotate, translate or otherwise move it (or ourselves) and look at it from different angles - introducing perspective. True, we must temporarily separate ourselves from our conceptual or physical starting point and move to another point of view, or to the object itself to do this (at which point we can use the ‘Smart Little People’ tool) and so, in many cases, create an apparent contradiction. However, as long as we remember to relate the multiple reference frames, via a strict set of angular proportions, we can turn the contradiction into analogy.

In the same way that mathematical or visual depth is created by juxtaposing opposing or angled points of view, the most refined methods of facilitating problem-solving attempt to lead the thinker through different angles of approach, to create psychological depth or insight.

Essentially, what contradiction implies, is that there is more than one reference frame or worldview in the equation. It implies that there is multi-dimensional, analogic or ‘9-windows’ information available to us, and not that there is actually a flaw in the system.

When contradiction is apparent, we are being called upon to identify the boundaries of our existing reference frame and move outside it. We are being asked to identify a geometric relationship and integrate the scale and perspective of our starting point, with the scale and orientation of our desired finishing point, so as to resolve the mismatch.

In essence, I am proposing that ‘multi-dimensional’, ‘systems-’ or inventive thinking is a process of resolving contradiction, to create a field of analogical, and hence triangulated, information. And, what is important in exploiting multiple levels of solution, is assuming a ubiquity of triangular relationship in nature and understanding their properties.

As problem-solvers, TRIZ allows us to add arbitrary elements and fields or dimensions, or remove them from the system, to enhance, decrease or remove undesired effects. We can do this in time or space and each time we do, the manipulation can be represented in a triangular relation.

Of course, mathematically we know that, because it can maintain both links and separation between opposites, the triangle is the most stable relationship for three elements.

More technically, it has no degrees of freedom. The triangle is totally deterministic, so a change in any one part will necessarily engender a change elsewhere in the triangle. Thus we can be sure of a result. If there is no obvious result, then we don’t have a triangle (though the converse isn’t true). By corollary, if we want to make a change in one part of a system without affecting another part, which might be vulnerable, we need at least 4 components -introducing a second level of triangulation.

It is well known that triangulation is the basic method of surveying, and that any topological problem can be broken down into a series of triangular relationships. Triangulation provides a ‘short-cut’, in that we can afford to ignore whole sets of data when we do our analysis, especially since we are projecting the problem onto 2D. The same method is used in computer-aided visualization of 3D surfaces and volumes [27]. Gross triangulations are made smoother by continually decreasing scale of triangulation. We can create a complete picture of our physical world in this way.

We could create a similar map of the changes in our psychological orientation and perspective during the process of problem-solving. A good artist could perhaps even apply the rules of scale and focussed perspective to the elements of this map, linking the original problem and coincident ‘primary’ triangulated reference frame of the observer, to all successive reference frames built onto it. At the conclusion, we would have quite a complex and certainly multi-dimensional representation of our solution on the page. Of course, in the Western tradition, we generally try to represent our problem in the simplest possible manner, so we never make pictures like this and it perhaps not obvious how multi-dimensional our thinking has been. However, as we resolve apparent contradiction of parameters via a process of analogy, we are always employing multi-dimensional thinking, whether we can see it or not.

In classical TRIZ, it is also known that the simplest model of a well-functioning technical system is the triangle, even though one of the elements (usually the field) may become hidden or ‘invisible’ when the desired effect is achieved. Triangulation can be exploited in Su-field analysis* and we might assume that contradiction is most easily resolvable when this tool is employed. Of course that is not exactly the case. However, with the existence of Su-field analysis, we can see that there is self-consistency within our toolbox; the concept of contradiction is not separate from the tools we use to resolve external contradictions, but intrinsic to them. In terms of this paper, we can say that the triangular relation is necessarily a TRIZ principle or Su-field principle because it is necessarily fundamental to the process of analogy.

In our daily lives and in technological innovation, we are often asked to deal with contradictions that Western scientific training does not well equip us for. In this paper, I have attempted to explore this issue by introducing analogy in its classical sense -not as a static entity or fait accompli, but as a process linking simple elements or dimensions up into more complex ones. Understanding that process allows us to find meaningful relationship where initially we see only irreconcilable difference.

In essence, I have proposed (1) The psychological concept of contradiction can be mathematically resolved, depicted and explained (2) The simplest geometric resolution of mismatched information occurs during triangulation. This is responsible for the creation of scale and orientation or depth and perspective in 2D and 3D -in our eyes, ears and minds. (3) The triangular relationship of elements, substances, fields, dimensions or reference frames is classically an analogic one.

I also propose that contradiction of any sort is best considered to be antithesis or paradox, rather than actuality. For, if we follow the steps of classical analogy to find proportional relationship between the elements, then the resolution of conflicting information or parameters is a means of achieving both stability and simplicity. In other words, recognizing that contradiction exists at various levels in our system is not considered to be a flaw or a problem per se, but a vital indicator that our preferred, primary thought system or physical orientation is at odds with respect to a second or nth reference frame and that we must find a relationship between them.

On the basis of the models presented in this paper, we can say that seeing contradiction is likely a fundamental part of the process of determining analogy -or creating fields of relationship. Conversely, the principle of triangular and analogic relation is the essence of contradiction both within and outside TRIZ, and the reason why TRIZ works so well.

In conclusion, we can say that the ideal final result of resolving contradiction is the process of analogy. And it would be exceedingly pleasant to also add that contradiction and analogy are fundamental, natural and universal rules of invention or decision-making -which is quite within the realms of possibility [28]… But of course, proof as to whether that is indeed the case must remain for another paper J.

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10. Eddington (1929) The Nature of the Physical World, Cambridge University Press, London.

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    Mann, D 2001 System Operator Tutorial (1) 9-Windows On The World, TRIZ-Journal, September

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    Mann, D 2001 System Operator Tutorial (4) Integrating Other Perspectives, TRIZ-Journal, December

14. For discussion of Fractal/fractional Dimension, Correlation Dimension, Hausdorff-Besicovich Dimension, Information Dimension, Generalized Dimension, Pointwise Dimension, Nearest-neighbour Dimension, Similarity- or Self-Similarity Dimension see: Berge P, Pomeau, Y and Vidal, C. 1984 Order within Chaos, John Wiley, NY, or Holzfuss, J and Mayer-Kress G 1986 An approach to error estimation in the application of dimension algorithms In Dimensions and Entropies in Chaotic systems, Mayer-Kress (Ed) Springer, Berlin pp 114-22 or Rasband, N.S 1990 Chaotic Dynamics of non-linear systems, Wiley NY.
15. Garnett P. Williams 1999 Chaos Theory Tamed, Joseph Henry Press, Washington
16. Professor Ross Hunt (1996) pers comm. Dept of Psychology, University of Western Australia. Refers to artist Leonardo Da Vinci and the physicist Sir David Wheatstone as being the earliest to recognize this though Brewster was the first scientist accredited with pointing it out…
17. Van der Wildt, A 1984 The Interchangeability of Space and Time in Perception In: Van Doorn, A., Van de Grind.,W.A and Koendernik, J.J. (Ed) pp139-172
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19. Van Doorn, A., Van de Grind., W.A and Koendernik, J.J. (Ed) 1984 Limits of Spatio-temporal correlation and the perception of visual movement In: Van Doorn, A., Van de Grind.,W.A and Koendernik, J.J. (Ed) 1984 VNU Science Press, Netherlands
20. See Hermann LF Helmholz (1954) On the Sensations of Tone: A Physiological Basis for the Theory of Music, Dover, NY
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23. Euclid (1660) Euclid’s Elements of Geometry; briefly, yet plainly demonstrated by Edmund Stone, Printed for Midwinter, Osborn and Longman, London.
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27. For up to date examples see perhaps proceedings of IEEE conferences at http://davinci.informatik.uni-kl.de/cgi-bin/vis98_build_tp_top or the online site for Centre of Computer Graphics and Data Visualization at the University of Bohemia http://herakles.zcu.cz/research.php
28. Anja-Karina Pahl (in prep) SYMPLicity: A SYstematic Method for seeing Pattern multipLICITY. From author’s PhD research at The University of Western Australia.
29. Julian Barbour (1989) Absolute or Relative Motion? : The Discovery of Dynamics Vol I. Cambridge University Press, London, or more simply, in Julian Barbour (2001) The End of Time Phoenix, UK.

An earlier version of this paper was presented at ETRIA conference (TRIZ Future) in Bath, UK, Nov 7-9, 2001. This version benefited greatly from discussion with the Biomimetics Group at the University of Bath -in particular Professor Julian Vincent and Dr. Olga Bogatyreva and from editorial comments of Dr. Ellen Domb and Dr. Michael S. Slocum at TRIZ-Journal.

The relation between a model and the thing to be modelled can be:

(a) Logical, in a formal system where there is structural similarity between model and system, and the same formal axiomatic and deductive relations connect elements and predicates of both system and model
(b) Replicative, in which there are material similarities between the parent system and its replica, and structural relations appear to be exactly reproduced, though in varying scale and degrees of detail, and/or in different media.

In either case, a member, x, of a set, is analogous to its fellow member, y, when either:

If x and y satisfy the first condition, they may be said to be substantially analogous (e.g. in the case of any two atoms). If the second condition holds, then x and y are formally analogous, irrespective of their constitution. If both conditions hold, the analogy is known as an homologous one.

Of course, homology implies both substantial and formal analogy exist, and substantial analogy implies formal analogy, but not conversely.

Further, if x and y are sets, then correspondence under condition (b) allows for several degrees of formal analogy:

(1) plain, or some-some analogy, when some elements of x are paired with some elements of y.

(2) injective, or all-some analogy, when every element of x is paired with an element of y.

If relations and operations (which may be defined as structures) in the injecting set are preserved and not modified, injective analogy is further defined as homomorphic.

(3) bijective, or all-all analogy, when the preceding relations hold both ways.

The special case of homomorphism from x into y and also from y into x, while the two morphisms compensate each other, is isomorphism. In modern terms, this is actually perfect analogy.

A dimension, in an everyday sense, is any measurable quality. It does not exist as an inherent feature of the cosmos. Rather, the creation of separated dimensions is an arbitrary, intellectual convenience, which originally comes to us courtesy of Euclid [23, 24] and Descartes [25, 26], and allows us to systematically, mathematically plot particular changes in the behaviour or qualities of the object or event of our interest on an imaginary graphical space. Usually, it is designed to separate an object of interest into discrete, measurable chunks and separate the result ‘done’ from the ‘doer’ and the ‘process of doing’.

However, we know a dimension, by its function, or what it does. Using only a static definition of dimension will keep us from seeing difference as intrinsic to establishing a relationship connecting identities or, conversely, to identifying the contradiction which breaks a harmonic relationship down.

Needless to say, factors which are hidden or implicit in our worldview are those which lead to paradox or apparent contradiction, when high levels of resolution are required. They can cause us to see lacunarity - glitches, gaps, other mismatches and apparent noise in our data, rather than coherent patterns.

Consider, for example, that we only know a straight line exists, by tracing it out from A to B in our minds or with a pen, in time as well as space. In Euclidean space, a straight line is defined as 1D. That means time (ostensibly a second dimension) must always be hidden, embedded or implicit in our measurement. Furthermore, it also means the observer (ostensibly a third dimension) must be hidden, embedded or implicit.

Interestingly, the line is only straight, if we collapse, suspend or remove time and project all the incremental reference frames created during the process of drawing onto a single piece of paper and consider this view is perpendicular to the ‘time axis’. Were we to take enough photos to completely cover the process of drawing our line on a flat page and then stack the individual frames, so that their bottom-left coordinates overlap with the origin of a Cartesian reference frame... and were we then to make an arbitrary, oblique slice through the stack, the cross-sectional view would not be straight but skewed or even arcuate. This is because when we stand at an oblique angle to our primary reference frame, we introduce perspective (an additional rotated reference frame) and apparently stretch space and time.

The graphs we prefer in the Western world usually have one, two or three mutually perpendicular straight line axes, labelled x, y and z, and allow us to speak of matter, energy and spacetime for example, as having 1D, 2D or 3D -as having respectively position, area and volume. These are our reference frame; a worldview or higher order starting point that results from combining one or more dimensions.

Regarding the function of a reference frame or worldview: It effectively allows contradictory variables to co-exist. It enables differences to be measured and depth to be created. If we say that each individual has a preferred worldview, called the primary reference frame, we can also say that that this may or may not be identical with the primary reference frame of the scientific society. Two or more reference frames may be related in analogic relationship, whereby new information can be immediately applied in apparently unrelated areas.

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